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SYSTEM DYNAMIC ANALYSIS OF A WIND TUNNEL MODEL WITH

APPLICATIONS TO IMPROVE AERODYNAMIC DATA QUALITY

A Dissertation submitted to the

Division of Research and Advanced Studies

of the University of Cincinnati

in partial fulfillment of the

requiremems for the degree of

DOCTOR OF PHILOSOPHY

in the Department of Mechanical,

Industrial, and Nuclear Engineering

of the College of Engineering

1997

by

Ralph David Buehrle

B.S.M.E., University of Akron 1985

M.S.M.E., University of Cincinnati 1988

Committee Chair: Randall J. Allemang, Ph.D.

https://ntrs.nasa.gov/search.jsp?R=19970015640 2020-07-30T15:53:42+00:00Z

SYSTEM DYNAMIC ANALYSIS OF A WIND TUNNEL MODEL WITH

APPLICATIONS TO IMPROVE AERODYNAMIC DATA QUALITY

ABSTRACT

The research investigates the effect of wind tunnel model system dynamics on measured

aerodynamic data. During wind tunnel tests designed to obtain lift and drag data, the

required aerodynamic measurements are the steady-state balance forces and moments,

pressures, and model attitude. However, the wind tunnel model system can be subjected

to unsteady aerodynamic and inertial loads which result in oscillatory translations and

angular rotations. The steady-state force balance and inertial model attitude

measurements are obtained by filtering and averaging data taken during conditions of

high model vibrations. The main goals of this research are to characterize the effects of

model system dynamics on the measured steady-state aerodynamic data and develop a

correction technique to compensate for dynamically induced errors. Equations of motion

are formulated for the dynamic response of the model system subjected to arbitrary

aerodynamic and inertial inputs. The resulting modal model is examined to study the

effects of the model system dynamic response on the aerodynamic data. In particular, the

equations of motion are used to describe the effect of dynamics on the inertial model

attitude, or angle of attack, measurement system that is used routinely at the NASA

Langley Research Center and other wind tunnel facilities throughout the world. This

activity was prompted by the inertial model attitude sensor response observed during high

levels of model vibration while testing in the National Transonic Facility at the NASA

Langley ResearchCenter.The inertial attitudesensorcannotdistinguishbetweenthe

gravitationalaccelerationandcentrifugalaccelerationsassociatedwith wind tunnelmodel

systemvibration,whichresultsin a modelattitudemeasurementbiaserror. Biaserrors

overanorderof magnitudegreaterthantherequireddeviceaccuracywerefound in the

inertial model attitudemeasurementsduring dynamictesting of two model systems.

Basedon a theoreticalmodalapproach,a methodusingmeasuredvibrationamplitudes

and measuredor calculatedmodalcharacteristicsof the model systemis developedto

correct for dynamicbias errors in the model attitudemeasurements.The correction

methodis verifiedthroughdynamicresponsetestsontwo modelsystemsandactualwind

tunneltestdata.

ACKNOWLEDGMENTS

I wish to thank my adviser, Dr. Randall J. Allemang, and my graduate review committee,

Dr. Dave Brown, and Dr. Robert Rost, for their guidance and support. ! am especially

grateful to Dr. Clarence P. Young, Jr. for his technical review of this document and his

advice during this research program. I thank my NASA supervisors, Mr. Richard A. Foss,

Dr. William F. Hunter, Dr. William S. Lassiter, Mr. Melvin H. Lucy, and Mr. William F.

Fernald, for their encouragement during my Ph.D. studies. ! would also like to thank Mrs.

Genevieve Dixon of the NASA Langley Research Center for assistance in the finite

element modeling area. Finally, I would like to thank my wife, Barbara, and children,

Bridget, Joseph and Blaine, for their patience and understanding during my studies. This

work was completed under the NASA research program, RTR 274-00-95-01, entitled

"Modal Correction for AOA Bias Errors".

TABLE OF CONTENTS

1.0 INTRODUCTION

1.1 Introduction

1.2 Problem Description

1.3 Literature Review

1.4 Solution Approach

2.0 EFFECTS OF MODEL DYNAMICS ON AERODYNAMIC DATA

2.1 Introduction

2.2 Force Balance Measurements

2.3 Transformation of Balance Forces

3.0 THEORETICAL FORMULATION

3.1 Introduction

3.2 Dynamic Equations of Motion

3.2.1 Lagrange's Equations

3.2.2 Kinetic Energy

3.2.3 Potential Energy

3.2.4 Energy Dissipation Function

3.2.5 Generalized Forces

3.2.6 Equations of Motion

3.3 Modal Analysis

3.4 Simplified Model

3.4.1 Two Degree of Freedom Example

I

2

11

16

18

19

19

25

25

28

28

29

30

30

31

31

34

35

4.0

4.1

4.2

4.3

2.4.2Extensionto MultipleDegreeof FreedomSystem

MODEL ATTITUDE BIASERRORCORRECTION

Introduction

ModalCorrectionTheory

ModalCorrectionImplementation

5.0 EXPERIMENTALVERIFICATION

5.1 Introduction

5.2 Wind-Off DynamicResponseTests

5.2.1 TestSetupandProcedure

5.2.2 CommercialTransportModelTestResults

5.2.3 High SpeedTransportModelTestResults

5.3 High SpeedTransportModelWindTunnelTests

5.3.1 TestSetupinWind Tunnel

5.3.2 DynamicResponseTestsin WindTunnel

5.3.3 WindTunnelTestResults

6.0 CONCLUDINGREMARKS

7.0 REFERENCES

APPENDIXA: Effectof AerodynamicForcesonModalCharacteristics

42

44

45

55

59

59

59

64

72

84

84

86

90

98

102

105

Figure 1.1

Figure1.2

Figure1.3

Figure1.4

Figure2.1

Figure2.2

Figure3.1

Figure3.2

Figure3.3

Figure3.4

Figure3.5

Figure4.1

Figure4.2

Figure4.3

Figure4.4

Figure5.1

Figure5.2

Figure5.3

Figure5.4

Figure5.5

Figure5.6

LIST OF FIGURES

National Transonic Facility model support system.

Schematic of wind tunnel model system.

Effect of vibration on inertial model attitude measurement.

Wind tunnel model instrumentation cavity.

Aerodynamic forces and model coordinate axes.

Influence of angle of attack error on drag coefficient for Cta=0.05.

Reference coordinate systems.

Sting bending in yaw plane, 9.0 Hz vibration mode.

Sting bending in pitch plane, 9.2 Hz vibration mode.

Two degree of freedom model.

Mode shapes for two degree of freedom example.

Harmonic motion of model at natural frequency of 0_y.

Yaw plane mode of model system.

Pitch plane mode of model system.

Flowchart of modal correction method.

Test setup in model assembly bay.

Shaker attachment for excitation in the yaw plane.

Sting bending in yaw plane, 10.3 Hz vibration mode.

Model yawing on balance, 14.4 Hz vibration mode.

Measured mean AOA, estimated bias, and corrected mean AOA

versus yaw moment for sinusoidal input at 10.3 Hz.

Measured mean AOA, estimated bias, and corrected mean AOA

versus yaw moment for sinusoidal input at 14.4 Hz.

4

5

9

10

20

23

26

36

37

38

41

47

53

54

57

6O

62

65

66

68

69

iii

Figure5.7 MeasuredmeanAOA, estimatedbias,andcorrectedmeanAOAversuspitchmomentfor sinusoidalinput at 11.2Hz.

Figure5.8 MeasuredmeanAOA, estimatedbias,andcorrectedmeanAOAversuspitchmomentfor sinusoidalinput at 16.2Hz.

Figure5.9 Stingbendingin yawplane,9.0Hz vibrationmode.

Figure5.10 Modelyawingonbalance,29.8Hz vibrationmode.

Figure5.11 InertialAOA measurement,yawacceleration,andyawmomentversustimefor 9.0Hz sinusoidalinputin yawplane.

Figure5.12 InertialAOA measurement,yawacceleration,andyawmomentversustimefor 29.8Hzsinusoidalinputin yawplane.

Figure5.13 MeasuredmeanAOA, estimatedbias,andcorrectedmeanAOAversusyawmomentfor sinusoidalinputat 9.0Hz.

Figure5.14 MeasuredmeanAOA, estimatedbias,andcorrectedmeanAOAversuspitchmomentfor sinusoidalinput at9.2Hz.

Figure5.15 (Top)MeasuredAOA andestimatedbiaserrorfor 9.2Hz sinusoidalexcitationin pitchwith 0.25Hzmodulation.(Bottom) Correspondingmeasuredbalancepitchmoment.

Figure5.16 (Top)MeasuredAOA andestimatedbiaserror for randomexcitationin pitch. (Bottom)Correspondingmeasuredbalancepitchmoment.

Figure5.17 Measuredandcorrectedmeanangle-ofattackfor sinusoidalexcitationat7.3Hz.

Figure5.18 (Top)MeasuredandcorrectedAOA for sinusoidalexcitationat7.3Hz with0.5Hz modulation.(Bottom)Correspondingbalanceyawmoment.

Figure5.19 Angle-of-Attack(AOA) for first sixty-foursecondsof awind tunneltestonahighspeedtransportmodel.

Figure5.20 (Top)Timedomainresponseof theAOA measuredwith theservo-accelerometersensorandthecorrectedAOA afterremovalof thedynamicallyinducedbiaserror. (Bottom) Correspondingtimedomainmeasurementof yawmoment.

70

71

74

75

76

77

79

80

82

83

88

89

91

93

iv

Figure5.21

Figure5.22

Figure5.23

(Top)Timedomainresponseof theAOA measuredwith theservo-accelerometersensorandthecorrectedAOA after removal of the

dynamically induced bias error. (Bottom) Corresponding time

domain measurement of yaw moment.

(Top) Time domain response of the AOA measured with the servo-accelerometer sensor and the corrected AOA after removal of the



(Top) Time domain response of the AOA measured with the servo-accelerometer sensor and the corrected AOA after removal of the



94

95

96

v

Table5.1

Table5.2

Table5.3

Table5.4

Table5.5

AppendixA Table1

AppendixA Table2

AppendixA Table3

LIST OF TABLES

Modal Parameters for Commercial Transport Model

Modal Parameters for High Speed Transport Model

Modal Parameters for High Speed Transport Model in Test Section

Summary of Wind Tunnel Results

Summary of Wind Tunnel Results for One Second Data

Acquisition Intervals

Transport model Worst Case Loading Conditions

Transport Model Natural Frequency Comparison

Transport Model Mode Radius Comparison

64

72

86

90

97

107

108

109

vi

aB(t)

a c

an(t)

at(t)

ax(t)

A fil

Ar

Aunf (t)

cij

CO

CFi

CC

CM

CM i

ccr

dcg/bc

dF/bc

di

D

LIST OF SYMBOLS

acceleration bias error estimate

centrifugal acceleration

time dependent normal acceleration

time dependent tangential acceleration

time dependent longitudinal acceleration

filtered AOA signal from inertial device

peak acceleration for r th mode

time dependent unfiltered AOA signal

damping coefficients

drag coefficient

force coefficient for degree of freedom i

lift coefficient

slope of lift coefficient versus angle of attack

pitching-moment coefficient

moment coefficient for degree of freedom i

correlation coefficient for least square fit of mode r

distance from model mass center of gravity to model balance center

distance from point of force application to model balance center

characteristic length corresponding to degree of freedom i

energy dissipation function

vii

f,

FA

Fc

FD

FL

FN

g

i

kr

m

m_j

mm

n

N

p_

Pry

qi

q_

natural frequency of r th mode in Hertz

axial force

centrifugal force

drag force

lift force

normal force

gravitational constant

scalar index

inertia about the y-axis for a reference at the model balance center

stiffness influence coefficients

bending stiffness

torsional stiffness

number of included modes

inertia coefficients

model mass

number of lumped masses used to represent the wind tunnel model system

number of degrees of freedom in the analytical model

modal coordinate for mode r

amplitude for mode ry

.thgeneralized coordinate

derivative of i th generalized coordinate with respect to time

dynamic pressure

viii

ai

an

OMi

?-

rp

ry

S

Si

T

t

U

Vr(t)

v,

V=

Yr(t)

rr

o_

O_

6

A

E

Pr

non-conservative generalized applied force (or moment) associated with qi

generalized aerodynamic force for translation degree of freedom i

generalized aerodynamic moment for translation degree of freedom i

current mode number

designates pitch plane mode

designates yaw plane mode

reference area of the model

reference area for degree of freedom i

kinetic energy of the system

time in seconds

potential energy of the system

time dependent velocity for rth mode

peak velocity for mode r

free-stream wind velocity

time dependent displacement for r th mode

peak displacement for rth mode

pitch rotation angle between undeflected sting and inertial coordinates

model attitude, or angle of attack

estimate of model attitude with bias error correction

difference

model attitude error

effective radius of r th vibration mode

ix

COl.

d

dt

_r

It]

[-,.]

[_

[_

(q}

{Q}

{Q'}

(p}

{_,}_

[_]

[_]

(x,y,z)

(xs, Ys, zs)

(x_o, Y_o, z_o)

(XBi, yBi, ZBi)

circular frequency of rth mode

derivative with respect to time

modal damping for mode r

matrix of damping coefficients

identity matrix

matrix of stiffness coefficients

matrix of inertia coefficients

vector of generalized coordinates

subset of generalized coordinates representing rigid model

vector of generalized forces

vector of generalized forces transformed to modal space

vector of modal coordinates

mass normalized modal vector for mode r

mass normalized modal matrix

diagonal matrix of natural frequency squared

modal, or eigenvector matrix

diagonalized damping matrix

inertial coordinate system

undeflected sting coordinate system

origin of sting coordinate system relative to the inertial coordinate system

body axis coordinate system for ith concentrated mass

x

(Xi, yi, Zi)

13)

position of ith body coordinate axes relative to sting axes

rotation of ith body coordinate axes relative to sting axes

AOA

BPF

c.g.

FEM

Hz

LPF

NTF

LIST OF ACRONYMS

Angle of Attack

bandpass filter

mass center of gravity

finite element model

Hertz

lowpass filter

National Transonic Facility

xi

Chapter 1

INTRODUCTION

1.1 Introduction

Model vibrations are a significant problem when testing in high pressure wind tunnels.

As discussed by Young [1], model vibrations can jeopardize model structural integrity,

overload force balances and support stings, cause models to foul, affect aerodynamic

data, and often limit test envelopes.

The National Transonic Facility [2] , NTF, is a transonic wind tunnel located at NASA

Langley Research Center which has the capability for testing models at Reynolds number

up to 140 million at Mach 1 and dynamic pressure up to 7000 pounds per square foot.

The NTF is a cryogenic facility with operating temperatures as low as -290°F. Severe

model vibrations have been encountered on a number of models since the tunnel began

operation in 1984. References 3 through 6 document studies of model and model support

vibrations in the facility. During a 1993 wind tunnel test, increased uncertainty in the

model attitude data was observed for periods of high model vibration. The response of

the onboard instrumentation to electrodynamic shaker input to the model without tunnel

airflow, "wind-off", was examined for two transport model systems [7, 8] at the NTF.

These wind-off dynamic tests found model vibration induced errors over an order of

magnitude greater than the required accuracy for the inertial model attitude

measurements.

This researchinvestigatestheeffectof wind tunnelmodelsystemdynamicsonmeasured

aerodynamicdata. The objective is to improve the aerodynamicdataquality during

conditions of high model vibrations.The equationsof motion are developedusing

Lagrange'sequationsfor the generalizedproblemof a cantileveredwind tunnel model.

This was the first time a systemdynamicanalysisapproachwas usedto examinethe

effectsof model vibrationson the aerodynamicdata. The modal solution of the

equationsof motion providesvaluableinsight into theunderlyingphysicsand provides

the basisfor the proposed"modalcorrectionmethod"for dynamicallyinducederrorsin

wind tunnel model attitudemeasurements.The proposedcorrectionmethodusesthe

modalpropertiesof themodelsystemto minimizethenumberof transducersrequiredfor

implementation. This is critical due to limited interior model spaceand thermal

considerationsassociatedwith cryogenicwind tunnelswhere heatedinstrumentation

packagesare required.The methodwas the first time domain techniquedevelopedto

compensatefor multiple modesin both the pitch andyaw planesof the modelsystem.

The ability to correct in the time domain is necessitatedby the randomnatureof the

measuredmodel dynamic responseand the increasedemphasison correlating time

dependentchangesin modelattitudewith aerodynamicloads.

1.2 Problem Description

The majority of wind tunnel tests are conducted with a model supported on the end of a

long tapered cylinder, referred to as a "sting", which is cantilevered from an arc sector or

movable vertical strut-type of support. A schematic of the NTF model support system is

2

shownin Figure 1.1. Pitchattitudeof themodelis adjustedby rotationof thearcsector.

Thearcsectorsystemis designedsuchthatthecenterof rotationof thearcsectoris at the

model,sothat changingthemodelpitchangledoesnot translatethemodelto adifferent

position relative to the wind tunnel test section. Roll attitudeof the model can be

adjustedby rotationof the sting. A six componentforcebalanceis usedasthe single

point of attachmentbetweenthe modeland supportstingas shownin Figure 1.2. In

order to achievethe desiredmeasurementaccuracyon three force and threemoment

aerodynamicloadcomponents,thebalanceis designedto be flexible ascomparedto the

sting. Theflexibility of thebalanceresultsin vibrationmodescharacterizedby themodel

vibratingasa rigid bodyona spring(forcebalance)in pitch,yawandroll. Thesemodes

are typically lightly dampedand often excitedduring wind tunnel testing [6]. Other

primarylow frequencyvibrationmodesareassociatedwith stingbendingin thepitch and

yaw planes,where most of the bendingdeformationoccursover the small diameter

portionof thestingnearthemodel.

This dissertationwill focuson the "pitch-pause"[9] wind tunneltest techniquesincethe

supportingwind tunnel testdatawere acquiredusingthis techniqueat the NTF. The

pitch-pausetechniqueis a commontestmethodusedto obtainaerodynamicloadsdatain

continuousflow, closedcircuit wind tunnels. In thepitch-pausetechnique,themodel is

movedto a prescribedangleof attackwith respectto the velocity vector,the transient

responsesareallowedto decay,andthentheforcebalance,pressure,andangleof attack

data are measured.At the NTF, the data measurementperiod is one second. The

3

Bearings

Sting

Model

Fixedfairing

Rolldrive

Locking pin -_

Arcsector

Crosshead

Shell

Insulation

Hydrauliccylinder

Figure 1.1 National Transonic Facility model support system.

Modelattach point

Balance momentcenter

X

Y

Balance/stingAOA Balance joint

package g sin(z

Figure 1.2 Schematic of wind tunnel model system.

5

procedure is then repeated for a series of model attitudes, which is referred to as a polar.

Increasing emphasis on wind tunnel productivity is pushing facilities towards shorter test

times (less time on point for transient dynamics to decay) and the effect on the

aerodynamic data accuracy must be evaluated.

During wind tunnel tests, free stream turbulence produce fluctuations in dynamic pressure

and flow angularity leading to unsteady forces on the model. The force balance and angle

of attack measurements are typically low-pass filtered and averaged to obtain "steady-

state" model attitude, aerodynamic force and moment data [10]. It is not unusual for the

peak-to-peak variation of the dynamic component of the "steady-state" force data to be

50% or more of the true mean. In Reference [11], the unsteadiness of the airflow and the

resulting model vibration is discussed. It is noted that, if the model vibration response

due to the flow unsteadiness is excessive, the ability to accurately measure the

aerodynamic quantities of interest may be compromised. Mabey [11] approaches the

problem by examining methods to reduce the flow unsteadiness in the wind tunnel. In

the Advisory Group for Aerospace Research and Development (AGARD) report entitled

"Wind Tunnel Flow Quality and Data Accuracy Requirements" [12], one of the data

accuracy issues is the measurement of, and correction for, aeroelastic deformations and

vibrations of models and support systems. Accuracy requirements [12] for lift, drag, and

pitching moment for transport type aircraft in the high speed regime are: Lift Coefficient

AC L = 0.01 ; Drag Coefficient AC D = 0.0001 ; Pitching-moment coefficient

AC M =0.001. In order to maintain the required accuracy, the tunnel free-stream

6

conditions must be repeatablewithin the following boundaries: Tunnel total and

stagnationpressure,AP= 0.1%;Model angleof attack,Ao_ = 0.01°; and Mach Number:

AM = 0.001. As an example, for conditions near a maximum lift to drag ratio, an increase

of 1 drag count (AC D = 0.0001) will decrease the payload by approximately 1% for the

long-range mission of a large transport aircraft.

The predominant instrumentation used to measure model attitude or angle of attack

(AOA) in wind tunnel testing at NASA Langley Research Center and wind tunnels

throughout the world is the servo accelerometer device described in Reference 13. The

inertial AOA package is shown installed in the nose of a test model in Figure 1.2. The

AOA package uses a servo accelerometer with its sensitive axis parallel with the

longitudinal axis of the model. For quasi-static conditions, this sensor provides a model

attitude measurement with respect to the local gravity field to an accuracy of _+0.01 ° over

a range of +20 °. An increment of 0.01 ° corresponds to an acceleration of 175 micro-g's.

During wind tunnel testing, the model mounted at the end of the sting experiences

dynamic oscillations due to unsteady flows that result in a bias error in the model attitude

measurement.

Young et. al [7] conducted an experimental study on the inertial model attitude sensor

response to a simulated dynamic environment in 1993 at the NTF. The experimental

study [7] clearly established that AOA bias error is due to centrifugal forces associated

with model vibration. For a single mode in simple harmonic motion, this is shown

7

schematicallyin Figure 1.3.TheAOA packagemovesonacirculararcabouta centerof

rotation that is mode dependent. For a single mode, the motion of the AOA package can

be treated similar to that of a simple pendulum. The centrifugal acceleration will act

outward from the center of rotation and be equal to the tangential velocity squared

divided by the radius arm. During wind-off dynamic tests, centrifugal acceleration due to

model vibration created a bias error over an order of magnitude greater than the desired

device accuracy of 0.01 degree. The bias error was found to be dependent on the

vibration mode and amplitude. The study revealed the complexity of the problem when

multiple vibration modes were present involving both pitch and yaw motions.

Although the Reference 7 study was conducted at the NTF, the AOA measurement error

due to model dynamics is not unique to this wind tunnel or to cryogenic wind tunnels.

The problem exists anytime model attitude is being measured by an inertial device in the

presence of significant model system vibrations. The amount of error in the inertial

model attitude measurement is dependent on the model system dynamics (i.e. will vary

for each model system) and is very difficult to quantify during actual wind tunnel tests.

Space limitations in wind tunnel models require that the number of additional transducers

used to implement a correction be minimized. This is illustrated by the wind tunnel

model instrumentation cavity shown in Figure 1.4. Also, in a cryogenic facility, such as

the NTF, special AOA sensor packages [13] are required. The instrumentation must be

placed in a heated package to maintain the sensors and obtain accurate and calibrated

AOApackage

a c

PF

(o r

oI. _.,

_d

X

Mode centerof rotation

Y

I___/

II

II

g a,I

II

III

I

X

Figure 1.3 Effect of vibration on inertial model attitude measurement.

C_

O

c_

o_ o

LT_

measurements at extreme temperatures (-290°F). Past experience with accelerometers

placed outside of the heated instrumentation package has revealed problems ranging from

sensitivity shifts due to temperatures variations to complete signal loss. The extreme

temperatures conditions, limited interior model space, and stringent accuracy

requirements necessitate placing the additional transducers necessary for correction of the

inertial AOA sensor output in the heated instrumentation package. The centrifugal

acceleration not only affects the inertial AOA device but can, if amplitudes are

sufficiently high, affect the desired axial force or drag measurement accuracy. The effect

of dynamics on pressure measurements can be a factor but is not addressed in this

dissertation.

1.3 Literature Review

Previous analyses of wind tunnel model system dynamics were restricted to a planar

problem. Burt and Uselton [14] examined the effects of sting vibrations on measured

dynamic stability derivatives. The equations of motion were derived for model rigid body

motion in the pitch plane using Newton's second law. Billingsley [15] uses Lagrange's

equations to derive the equations of motion for a cantilevered sting-model system. Again,

the derivation is restricted to motion in the pitch plane. Young et. al [7] have shown that

model yaw vibration can result in an error in the measured pitch angle for a model-

mounted inertial angle of attack device. Therefore, an analytical model is required that

includes both pitch and yaw plane dynamics to better evaluate the effects of model

dynamics on the measured aerodynamic data.

11

The first correctiontechniquefor modelvibrationinducederrorsin inertial wind tunnel

modelattitudemeasurementswasdevelopedin 1984by PeiterFuijkschotof theNational

AerospaceLaboratory in the Netherlands[16]. This time domain technique was

developedfor one vibration modein eachthe yaw and pitch plane. Two additional

accelerometersareusedto measurethetangentialaccelerationsdueto the yawandpitch

motion of the model. The tangentialaccelerationsare integratedto obtain velocity,

squared,and divided by a scalefactor to compensatefor the effective radius of the

vibrationmode. Thissignalis thenaddedto theunfilteredAOA outputto cancelthebias

term.Themoderadiusin theyawandpitchplaneis determinedby tuningapotentiometer

while manuallyexciting the model in the yaw and pitch plane,respectively. A major

drawbackis that this techniquedoesnot addressthecasewheremultipleyaw andpitch

modesarepresent.

Renewedinterest in the effects of model vibrationson the measuredaerodynamic

quantities was prompted by the 1993 study of Young et. al. [7]. Prior to this

investigation,only asinglemodein themodelpitchandyawplaneswasconsidered.This

studyshowedthepotentialfor multiplemodesin eachplaneto participate.Severalrecent

studieshavebeenconductedatNASA LangleyResearchCenterto examinetheeffectsof

modelvibration on model attitudemeasurementdevices[7, 8, 17, 18]. In addition,

analysisof the vibration effects on gravity sensinginclinometers is underway by

Fuijkschot[19,20]of theNationalAerospaceLaboratoryin theNetherlands.

12

Frequencydomaincorrectiontechniqueshavebeenproposedby Young et. al [7] and

Tchenget. al [18]. Thecorrectionmethodof Younget. al is derivedusingan average

displacementof the model throughone cycleof vibration. This methodrequiresthe

measurementof thenaturalfrequenciesandcorrespondingpeakaccelerationmagnitudes

from thefrequencyspectraof theyawandpitchaccelerations.Youngproposesthat the

requiredscalefactor,effectiveradius,bedeterminedempiricallyduringwind-off ground

vibration tests.The correctionmethodof Tcheng[18] requiresthe measurementof the

natural frequenciesfrom the frequencyspectraof the tangentialaccelerationsand the

secondharmoniccomponentsfrom thefrequencyspectrumof theunfilteredAOA signal.

This techniqueis difficult to implementdueto theparticipationof multiple modesand

the required data accuracyto measuresmall magnitudesat the secondharmonic

frequency.Bothtechniqueshaveimplementationproblemsdueto therequiredfrequency

domainsignalprocessingof randomwind tunnel test dataover short (1 second)data

acquisitionperiods.

Another method under developmentby Tripp [8] usestime and frequencydomain

analysesto estimateand correct for the dynamicbias error. The proposedtime and

frequencydomainbiaserror correctionalgorithmis basedon thebias term for a single

yawmodebeingrepresentedby thesquareof thevelocitydividedby themoderadius. A

sensitivecorrelationtestbetweentime seriesis providedby the crossspectraldensity

coherencefunction. Correlatedspectralcomponentscommonto boththeunfilteredAOA

signalandsquareof the dynamicyawor pitchmeasurementappearin the crossspectral

density coherencefunction. Other spectralcomponentscommonto the auto spectra

13

which arenot phasecoherent,i.e. unsynchronized,tendto be removedfrom the cross

spectrumby averagingand canceledby normalization,and do not appearin the cross

spectralcoherencefunction. Thecoherencefunctionandcrossspectrumthusprovidea

meansof detectingandquantifyingAOA biaserrorsdueto angularoscillation. Thecross

spectraldensitycoherencefunctionis examinedfor spectralcorrelationwithin the AOA

passbandand the correspondingmodal frequenciesare identified. The modal radius

correspondingto eachnaturalfrequencyis estimatedby aleastsquaresfit of the integral-

squaredyaw(or pitch) measurementto thedynamicAOA output.This requiresa longer

datarecordinitially (>_10seconds)to obtaina goodestimateof the moderadius. This

moderadiusis thenusedasaconstantfor theremainderof thedatapoints. A bandpass

filter aboutthemodalfrequencyis usedto isolateaparticularmode.Theresultingsignal

is thennumericallyintegratedandsquaredanddividedby the scalarmoderadiusto give

thebiaserrorassociatedwith a particularmode. ThecorrectAOA output is thenfound

by subtractingoff the contributionsfrom all of the modesshown to have spectral

correlationandlow-passfiltering the result. In wind-off dynamictests[8], this method

had implementationproblemsdueto significantlow frequencyrandomdisturbancesin

the integral-squaredyaw (or pitch)measurementswhich wereabsentin the AOA time

series.

After the needto compensatefor multiple vibrationmodeswasdemonstratedat NASA

Langley ResearchCenter [7,8,17], Fuijkshot extendedhis time domain correction

techniqueto compensatefor multiple modes[19, 20] usingEuclideankinematicsof a

14

solid body. This work was donein parallel with the proposedtime domain "modal

correctionmethod"that is thesubjectof this dissertation.For a givenplaneof motion,

Fuijkshotproposesmeasuringboth therotationalrateandthevelocityof therigid model

anddeterminingthecorrectiontermfrom theproductof the two signals. The rotational

rateandvelocity signalsfor theyaw(orpitch)planewill containthecontributionsfor all

modesacting in that plane. The radiusfor eachof the modeswill not need to be

determinedexplicitly. Thecorrectiontermsfor thepitchandyawplanesarethenadded

to theunfilteredAOA signalprior to filtering. Themethodis currentlyunderevaluation

andhasbeenverified for sinusoidaltests[20]. Thevelocitycanbedeterminedthrough

integration of an accelerometersignal. In application,the rotational rate has been

obtainedby integratingthe differencefrom two linear accelerometersattachedto the

model fuselage,oriented in the yaw (or pitch) plane, divided by the accelerometer

separationdistance. This assumesthe accelerometersareconnectedby a rigid model

fuselage. This correctiontechniquerequiresfour additional transducersin order to

determinethe rotational rate and velocity in the pitch and yaw planes. The limited

interior spacein modelsandextremetemperatureenvironmentsin somewind tunnels,

where heatedinstrumentationpackagesare required, may prohibit this number of

transducers.This methoddoesnot providea meansof checkingthe rigid-bodymodel

assumptionsuponwhichit isbased.

In general,theproposedtime domaincorrectionsprovideseveraladvantages.First, time

domainsignalprocessingcanbe appliedto the randomwind tunnel testdataacquired

15

over shortdatasamplingperiods. Secondly,the inertial AOA packageoutput can be

correctedfor the dynamicallyinducederrorsto give an accuratetime domain model

attitudesignal. The measurementof time varying signalsand analysisof this data is

becoming a more significant requirementfor subsonicand transonic experimental

researchers[21].Themeasurementof instantaneousandaveragevaluesof modelattitude

andcorrelationwith measuredmodelloadsis gainingincreasedinterest.

1.4 Solution Approach

The research is divided into the following four areas: examination of the effects of

models dynamics on aerodynamic data; development of a theoretical model; development

of a correction for model vibration induced errors in inertial wind tunnel model attitude

measurements; and experimental verification.

In Chapter 2, the significance of the problem is shown by examining the effects of model

dynamics on the measured drag force and corresponding drag coefficient. Errors

introduced by the centrifugal forces associated with model vibration are quantified. The

propagation of the angle of attack errors during the transformation of the measured forces

from the model body axes to the wind axes is also examined.

In Chapter 3, the governing equations of motion for a cantilevered wind tunnel model

system are derived in discrete form using Lagrange's equations. This formulation

describes both pitch and yaw plane dynamics. The equations of motion are solved using a

16

modalanalysisapproachto obtainthe generalized,modal,solution.Basedon observed

behaviorof wind tunnelmodelsystems,theproblemis simplified.

In Chapter4, thetheoreticalmodelis usedto developa time domaincorrectionmethod

for modelvibrationinducederrorsin inertialwind tunnelmodelattitudemeasurements.

The implementationof the proposed"modal correctionmethod" using digital signal

processingtechniquesis alsodescribed.

In Chapter5, themodalcorrectionmethodis verifiedthroughacombinationof wind-off

dynamictestson two transportmodel systemsand wind tunnel testdata. The modal

correctionmethodis appliedto wind-off modeldynamicresponsedata for sinusoidal,

modulatedsinusoidalandrandomshakerinputsin thepitchandyawplane. In addition,

the modal correctionmethodis appliedto measureddynamicresponsedata recorded

duringwind tunneltestingof atransportmodelin theNTF.

In Chapter6, theresearchresultsaresummarizedandrecommendationsfor futurework

aredescribed.

17

Chapter 2

EFFECTS OF MODEL DYNAMICS ON AERODYNAMIC DATA

2.1 Introduction

For wind tunnel data acquisition, Steinle and Stanewsky [12] recommend that samples of

data be taken over a time interval sufficient to average out the effects of dynamic

response and unsteady flow to establish the desired confidence interval. However, as

discussed by Buehrle and Young [17], the centrifugal acceleration created by model

vibration results in a bias error in the inertial wind tunnel model attitude measurement.

For wind-off sinusoidal model response, it is shown that the inertial angle of attack

measurement has a mean offset which cannot be removed by filtering or averaging.

Errors over an order of magnitude greater than the required device accuracy of 0.01 ° are

possible [7, 8].

In this chapter, the effect of model vibration on the force balance measurements is

quantified. The direct effect of the vibration induced centrifugal force on the accuracy of

the measured forces is examined. Typically, the forces and moments are measured by an

internally mounted strain-gage balance which has a coordinate system that is fixed to the

model. This data is transformed to obtain the desired lift and drag force components

using the measured model attitude. The propagation of the model attitude error during

the transformation process is also examined.

18

2.2 Force Balance Measurements

Model vibration induced centrifugal forces result in errors being introduced into the

balance forces. The centrifugal force F can be writtenC

Fc = mma c (2.1)

where m m is the model mass and a is the total centrifugal acceleration. It is anticipatedC

that the centrifugal force acting on the model will be small. For the servo accelerometer,

a centrifugal acceleration of .00175 g's corresponds to a model attitude error of 0.1

degrees, which is 10 times the required device accuracy. This same centrifugal

acceleration will result in only a 0.26 pound centrifugal force for a model weighing 150

pounds. For the high dynamic pressure wind tunnel tests that produce significant model

dynamics, this would result in a drag coefficient error less than the required accuracy.

2.3 Transformation of Balance Forces

A more significant error in the measured forces may occur due to errors in the measured

model attitude. The propagation of the model attitude error into the measured drag force

was described by Owen et. al. [22]. The strain gage balance forces are measured in the

model body axes, which are fixed to the model, and transformed to the lift and drag force

components using the measured model attitude. Figure 2.1 shows the relevant forces and

coordinate axes. The axial force, FA, and normal force, FN, are the balance forces

measured relative to the body axes, (XB, ZB). The lift force, FL, and drag force, FD, are

defined relative to the wind axes, (x ,z), which have one axis parallel to the flow

direction. The measured model attitude, _, defines the transformation between the two

19

F'ltFN

direction

ZB ¢ _rz L_ FA

Figure 2.1 Aerodynamic forces and model coordinate axes.

20

coordinate systems. The lift and drag forces can be written

F L = F N cos(Or ) - F A sin(or )

FD = F a cos(ct ) + F N sin(or )

If the model attitude has an error, E, the lift and drag forces can be written

FL = F N cos(o_ + _ ) - Fa sin(or + E )

Fo = F a cos(or + E ) + FN sin(oc + Iz)

(2.2a)

(2.2b)

(2.3a)

(2.3b)

The errors in the lift and drag forces due to the model attitude error, e, are defined by the

differences of Equations 2.2 and 2.3.

AF L = FN (cos0x + E ) - cos(o_ )) - FA (sin(o_ + E ) - sin(or )) (2.4a)

AF D = FA (cos(or + _ ) - cos(o_ )) + FN (sin(or + e ) - sin(o_ )) (2.4b)

Expanding the trigonometric expressions and applying small angle assumptions for e,

AF L = -e (F N sin(or )+ FA cos(o_ ))

AF D = _ ( FN cos(or ) - Fa sin(cx ))

Substituting from Equation 2.2 for the terms in parentheses results in

AF L = -EF o

AF D = eE L

The aerodynamic forces are expressed in coefficient form as

(2.5a)

(2.5b)

(2.6a)

(2.6b)

C L - EL and C D - FD (2.7)q_S q_S

where CL is the coefficient of lift, CD is the coefficient of drag, q_ is the dynamic

pressure, and S is the reference area of the model.

21

RewritingEquation2.6 in coefficientform gives

AC L = -EC D

ZXCD= ECL

(2.8a)

(2.8b)

As discussed in Chapter 1, the accuracy requirements [12] for lift and drag measurements

for transport-type aircraft in the high speed regime are: Lift Coefficient, AC L = 0.01 ;

Drag Coefficient, AC D = 0.0001. Except for conditions near zero lift, the coefficient of

drag is significantly less than the coefficient of lift [23]. Therefore, the error in drag

coefficient will be more critical with regard to its required measurement accuracy.

Assuming the lift coefficient can be represented as a linear function of model attitude,

gives

CL = CL_ O_+ Cons tan t (2.9)

where CLa is the slope of the lift coefficient versus model attitude plot. Substituting the

results of Equation 2.9 into Equation 2.8b gives

AC D = rt180 e (CLa °t + Constan t) (2.10)

where ot and E are expressed in degrees. The slope of the lift coefficient versus model

attitude plot for several characteristic wing shapes range from 0.05 to 0.1 per degree [23].

Using the most conservative, lower, value of CLa = 0.05per degree, the error in drag

coefficient versus model attitude is plotted for several values of model attitude error in

Figure 2.2. For this plot, the constant term in Equation 2.10 is set to zero. For a non-

zero constant term the lines will be shifted, however, the basic trends will be consistent

with those shown.

22

I,g

.eu

iE0

C3

0.0025

0.002

0.0015

0.001

0.0005

0

0

.... AOA Error = 0.20

.... AOA Error = 0.15

...... AOA Error = 0.10

- - -- AOA Error = 0.05

--AOA Error = 0.01

J

,f,i

f

,I

Jf ,i"

°f J° °i¢J '°I°

j° I

• J .I

j° °_° °J I °'°

jJ .I ° ..o-"

j° _' .°f .I" o''"

J ._ .-°°

J

2 4 6 8 10 12

Angle of Attack (Degrees)

Figure 2.2 Influence of angle of attack error on drag coefficient for CLo _=0.05.

23

As can be seen from Figure 2.2, significant errors in drag coefficient can occur due to the

propagation of the errors in the model attitude measurement. Model attitude errors

equivalent to those measured in wind-off dynamic tests (E>0.1 o) [7, 8] would result in an

error in the drag coefficient that is an order of magnitude greater than the required

accuracy at high angles of attack.

24

Chapter 3

THEORETICAL FORMULATION

3.1 Introduction

In this chapter, the equations of motion for a cantilevered wind tunnel model system are

derived using Lagrange's Equations [24- 26]. The Lagrange method provides a

generalized systematic energy approach for defining the equations of motion in any

convenient coordinate system. The resulting equations of motion are formulated in terms

of the generalized, modal, coordinates. Based on observed behavior of the model system

during wind tunnel tests, the analytical model is simplified. This simplified model

provides the basis for development of the modal correction method in Chapter 4.

3.2 Dynamic Equations of Motion

A lumped mass model will be used to represent the wind tunnel model and its support

system. This work extends the planar analysis of Billingsley [15] to include both pitch

and yaw dynamics of the sting-balance-model system.

In order to represent the model system during pitch-pause wind tunnel testing, three

coordinate systems are defined in Figure 3.1. The coordinate system (x, y, z) is the

inertial coordinate system with the x-axis parallel to the wind direction. The coordinate

system (xs, ys, z_) is fixed to the undeflected sting axis and has its origin at the arc sector

center of rotation. Recall from Chapter 1, that the arc sector is the movable portion of the

model support system that provides the pitch adjustment for the model. The arc sector is

25

° ,,,,I

X

N

designed such that its center of rotation is at the model, so that changing the model pitch

angle does not translate the model to a different position relative to the wind tunnel test

section. The coordinate system (x_, Ys, z_) can be defined by the location of its origin (xs0,

ys0, Zs0) relative to the inertial coordinate system and the pitch angle (_). The body axes

(xBi, yBi, zBi) are fixed to the ith concentrated mass. The position and orientation of the

body axes for the ith concentrated mass relative to the undeflected sting axes are defined

by the translations (Xi, Yi, Zi) and rotations (7i, oq, _i).

In the pitch-pause method of wind tunnel testing, the model is pitched to a desired angle

(eq) and paused to establish "steady-state" conditions. This results in:

Xs0 = J's0 = Zs0 = _s = 0 (3.1)

where the "o" denotes the derivative with respect to time. The time varying components

representing the motion of the ith mass are the translations (xi, yi, zi) and rotations (Ti, oq,

_i) relative to the undeflected sting. The generalized coordinates describing the motion of

the cantilevered sting-balance-model system can be written:

{q} ={Xl Yl Zl T1 °_ 1 _1 ... Xn Yn Zn '_n °_n _n} T (3.2)

where n is the number of lumped masses used to represent the wind tunnel model system.

The rotation angles (7i, _, 13i) induced by inertial and aerodynamic loading are small.

Therefore, in the subsequent derivations, small angle approximations (i.e., sin(o_ )----o_;

cos(o_ ) = 1) can be used and higher order terms can be neglected.

27

3.2.1 Lagrange's Equations

For a lumped mass model, Lagrange's Equations [26] can be written as:

d ( _T "_ _T _D _U

-_--j-_-+--+--=Qil[_qi_qi _qi _qi for i=l ..... N

where:

(3.3)

D = energy dissipation function

n = number of lumped masses used to represent the wind tunnel model system

N = 6*n = number of degrees of freedom

qi = ith generalized coordinate

t_i = derivative of ith generalized coordinate with respect to time

Qi = non-conservative generalized applied force (or moment) associated with qi

T = kinetic energy of the system

U = potential energy of the system

3.2.2 Kinetic Energy

The kinetic energy of the system can be written [26]:

1 NN

T = "_i_=l _. mijqiqj (3.4)J=l

where the mij are inertia coefficients. For small oscillations about the equilibrium, the

inertia coefficients are constants and the kinetic energy is a function of {q} only. The

mass matrix is symmetric, i.e., mij=mji .

_T_=0

3qi

Since the kinetic energy is not a function of {q},

(3.5)

28

Taking the derivative of the kinetic energy with respect to the time derivative of the ith

generalized coordinate gives:

OT N_ Y_mijitj (3.6)

Oqi j=l

The time derivative of Equation 3.6 is:

.-_tt(,-_qi )= jE=lmiSl j (3.7)

3.2.3 Potential Energy

For the cantilevered wind tunnel model, the potential energy is equal to the strain energy

stored in the sting-model system. A detailed derivation of the strain energy is given by

Fung [27]. The potential energy can be written in terms of the stiffness influence

coefficients as:

NN

l i_=lj_=lkt.lqtq j (3.8)U 2_-- .. . .

where the stiffness influence coefficient, kij, is the force required at point (i) due to a unit

deflection at point (j) with all other points held fixed. The stiffness influence coefficients

are symmetric, i.e., kij = kji. Taking the derivative of the potential energy function with

respect to the generalized coordinate ( qi ) gives:

o_U N

_ _., kijq j (3.9)Oqi j=l

29

3.2.4 Energy Dissipation Function

For the case of viscous damping, a dissipation function, D,

energy function can be defined [26].

N N

where the damping coefficients, cii, are symmetric, i.e., cij =cii.

the dissipation function with respect to the time derivative

coordinate gives:

OD N- Z co0 j

_)t)i j=l

analogous to the potential

(3.10)

Taking the derivative of

of the ith generalized

(3.11)

3.2.5 Generalized Forces

The primary generalized forces are the unsteady aerodynamic loads. The aerodynamic

loads will be modeled using a quasi-steady approximation [15]. The generalized

aerodynamic forces associated with the translation degrees of freedom are modeled as:

QFi = q_,,SiCF i (3.12)

where, q. is the dynamic pressure and Si is the characteristic area. The coefficient CFi

will be assumed linear and is a function of the model attitude. Similarly, the generalized

aerodynamic moments associated with the rotational degrees of freedom are modeled as:

QM i = q_SidiCM i (3.13)

where di is a characteristic length and the coefficient CMi is assumed to be a linear

function of model attitude.

30

3.2.6 Equations of Motion

The equations of motion for the ith lumped mass can be obtained by substituting the

results from Equations 3.5, 3.7, 3.9, 3.11, 3.12 and 3.13 into Equation 3.3. In matrix

form this yields:

[ M]{/_} + [C]{q} + [K]{q} = {Q} (3.14)

where,

{q} ={Xl Yl Zl 3'1 °_ 1 Ill ... Xn Yn Zn "_n O_n _n} T

[C] is a square matrix of the damping coefficients, cij

[K] is a square matrix of the stiffness coefficients, kij

[M] is a square matrix of the inertia coefficients, mij

{a} is a vector containing the generalized forces, Qi

3.3 Modal Analysis

The modal analysis technique [24, 28] will be used to solve for the dynamic response of

the multiple degree of freedom system described by Equation 3.14 with initial conditions

{q(0)}={q0} and {q(0)}={q0}. The modal analysis technique is based on the

transformation of the coupled equations of motion represented by Equation 3.14 into an

independent set of equations using the normal modes of the system.

In the modal analysis technique, the first step is to obtain the eigenvalues and

eigenvectors associated with the mass and stiffness matrices of the system. Numerical

31

methodsfor solving theeigenvalueproblemarediscussedin References21, 23 and 26.

Another approachis to obtain the eigenvaluesand eigenvectorsthroughexperimental

modalanalysis[29,30]. Oncethenaturalfrequenciesandmodeshapesareobtained,the

solutionto theeigenvalueproblemcanbewrittenas:

[M][_]['032.]= [K][W] (3.15)

where, [W] is themodalor eigenvectormatrix

['032] is adiagonalmatrixof thenaturalfrequencies,_ ,squared

Normalizingthemodalmatrixwith respectto themassmatrixyields:

[_]T [M][_] = ['i. ] (3.16a)

[_]T [K][_] = ['032.] (3.16b)

where,[_] is themassnormalizedmodalor eigenvectormatrix,and

['I. ] is the identitymatrix

Thetransformationfrom thegeneralizedcoordinates,{q}, to themodalcoordinates,{p},

canbewritten:

N

{q(t)}=[_]{p(t)}= ]_ {d_}rPr(t) (3.17)r=l

where, {_ }r is the mass normalized modal vector for mode r. Substituting Equation

3.17 into Equation 3.14, and premultiplying by [_]T yields,

t.Ftdt.l{/,}+[,02.]{p}=t.F

32

Assuming the damping is a linear combination of the mass and stiffness matrices, the

transformation will also diagonalize the damping matrix.

[CI_]T [C][cI_] = ['2403. ] (3.19)

where the modal damping for mode r can be written:

1 {_}T[c]{ _}r (3.20)_r -- 20_r

Substituting Equation 3.19 into 3.18 results in

+ ]{p}+ ]{p}--{Q,}

where

(3.21)

{Q,}_-E_,Ir{Q} (3.22)

The N independent equations corresponding to Equation 3.21 can be written as

_0r (t) + 2 4 rO_rP(t) + O_2 p(t) = Qr (t) , r=l,2 ..... N (3.23)

This is the form of a single degree of freedom system with viscous damping. Using the

transformation equation (3.17), the initial conditions can be written

{q(0)}= [_]{p(0)} and {q(0)}= [_]{p(0)} (3.24)

Premultiplying these equations by [_]T[M] and solving for the modal initial conditions

gives

pr(0) = {_ }T[M]{q(0)} and/Sr(0) = {_ }T[M]{q(0)}, for r=l,2 .... ,N (3.25)

33

The solution to Equation 3.23 can be obtained using the Laplace transform method [24].

This results in

pr(t) = 1--_-i Qr(X )e -_rO3r(t-'_ ) sin0_dr (t -Z )dxO_d r 0

+e r rtl r O cos dsin dr/ J(3.26)

where (Ddr = O r _/(1- _ r2 ) is the damped natural frequency for mode r.

For a given set of generalized forces and initial conditions, Equations 3.22, 3.25 and 3.26

can be used to solve for the modal coordinates, {p}. The solution in terms of the

generalized coordinates, {q}, can then be found from Equation 3.17. The problem is

now in generalized form and can be used to estimate and correct for model vibration

induced centrifugal accelerations. However, the problem can be simplified as developed

in the following section.

3.4 Simplified Model

Once the natural frequencies and mode shapes have been obtained, the dynamic model of

the sting-model system can be simplified based on behavior observed during wind tunnel

testing. The primary dynamic components affecting the wind tunnel model

instrumentation are in the model pitch and yaw planes [7, 8]. Since the inertial angle of

attack device has its sensitive axis parallel to the longitudinal axis of the model, the

34

deviceis notsensitiveto roll motionsaboutthisaxis. Also, theeffectsof axialmodeson

the inertial angle of attack device can be removed through filtering. Therefore, only the

pitch and yaw plane motions will be considered in subsequent derivations.

Figures 3.2 and 3.3 show measured mode shapes of a high speed commercial transport

model in the National Transonic Facility (NTF). These mode shapes demonstrate several

important characteristics common to models tested in the NTF. The lower frequency

modes (<50 Hz) of the model system are characterized by rigid body motion of the model

on the more flexible sting-balance combination. The first two modes are associated with

sting bending motion in the pitch and yaw plane. In order to achieve the desired

measurement accuracy for the "steady-state" aerodynamic loads, the force balance is

relatively flexible as compared to the model and sting. The strain gage balance systems

used in the NTF [31] are designed with flexures that separate the loads into its planar

components with minimal interactions. This results in predominantly pitch or yaw plane

motion of the model for the lower frequency modes of the system. For a given mode, the

rigid-body model motion can be defined by a translation y or z along with a

corresponding rotation [3 or t_ (see Figures 3.2 and 3.3).

3.4.1 Two Degree of Freedom Example

A two degree of freedom example will be used to define some useful properties

associated with the planar motion of the "rigid" model. The modal characteristics of the

two degree of freedom system shown in Figure 3.4 will be examined. This is similar to an

35

0

E_E

_" •a

I

>

/E_c--

0

0

N

0

°_,,q

.8

t"l

o E

._ I

IIII

N

,/

X

_h.._r

0

0

N

._..q

J_

r_

._,.qLT._

KT

KBBalance moment

center

O_

Figure 3.4 Two degree of freedom model.

38

example for vehicle suspension given by Thompson [32]. In this example, the translation

and rotation coordinates at the balance moment center will be used in defining the model

motion. Using the Lagrange Method, the equations of motion are derived. For small

angles, the equations of motion are

I-mm'mdmcg/bc -mrn'dcglbc-_Z_+IkBIybcJ_(iJ [. 0 kOT_Z}=fdF/lbc} f(t) (3.27)

where, mm is the model mass, dcg/bc is the distance from the mass center to the balance

center; dF_ is the distance from the force to the balance center; Iy bc is the inertia about

the balance center; kB is the bending stiffness; kT is the torsional stiffness; z and o_ are the

displacement and rotation from the equilibrium position; and F(t) is the applied force.

The main interest is in the form of the mode shapes. The eigenvalue problem

corresponding to Equation 3.27 can be written

0 z mrn mrn'dcg/bc_Z l (3.28)

Based on measured weight, physical dimensions, and natural frequencies of a typical

transport model system, the following constants were determined.

mm=0.3313 pound-secondZ/inch

Iy bc= 19.51 inch-pound-second 2

dc_c= 5 inches

kB = 1308 pound/inch

kT = 277089 inch-pound

39

SubstitutingthesevaluesintoEquation3.28,andsolvingtheeigenvalueproblemyields

°_1 _9.38Hz;{_}1 1-1"513lfl - 2*rt = [0.0415J

(3.29a)

°_2 -26.7Hz" {q_}2 l 1"719 lf2 - 2*n ' = 10.2955J

(3.29b)

The mode shapes are depicted graphically in Figure 3.5. Note that for each mode there is

a node (point of zero motion) about which the rigid body model rotates. The position of

this node is defined by the ratio of the translation and rotation degrees of freedom.

Scaling the modes to unit rotation gives

{q_}l :0"0415 l 1 I =

[5.82] 0.2955{- _2 }

(3.30a)

(3.30b)

where the ith mode radius, Pi, is defined as the ratio of the translation and rotation mode

shape coefficients with the modal vector scaled to unit rotation. This yields a physical

interpretation of the mode radius as the distance from the node to the reference point on

the model with the positive direction defined by the model x-axis. For this example, the

mode radius values are Pl = 36.4 inches, and P2 = -5.82 inches. The radius by definition

can be positive or negative based on the mode shape. The effect of the sign of the radius

will be discussed in Chapter 5.

40

Mode 1

_ _ Node

.............. _ -_- x

Mode 2

P2 = -5.82 inch

Z

Figure 3.5 Mode shapes for two degree of freedom example.

41

3.4.2 Extension to Multiple Degree of Freedom System

The results of the two degree of freedom example can be used to simplify the

transformation Equation 3.17. Recognizing the planar characteristics of the model

response for the lower frequency modes, Equation 3.17 can be expanded as

{q(t)}:[dP]{p(t)}= _.{f) }ryPry(t)+ Y,{_ }rpPrp(t)+ _, {t_ }rPr(t)ry rp r_ry ,rp

(3.31)

The low frequency yaw modes denoted by ry are characterized by rigid body motion of the

model. Letting {q} be the subset of the generalized coordinates required to represent the

model fuselage, yields:

{q}ry = _ry{_f}ryPry (t)

!o

0

ry 00

(t) = _.t_ _BryPryry

ry

0

- Pry

0

0

0

1Pry (t)

(3.32)

The coordinates shown represent the x, y, z, T, or, and [3 degrees of freedom for a point on

the "rigid" model fuselage.

Similarly, for the low frequency pitch plane modes, rp , the rigid body motion of the

model is approximated by

rp

!o0rEp -PrpPrp(t)= t_a rp 0

1

0

Prp (t) (3.33)

42

For a given mode,the rotationand translationdegreesof freedomin the predominant

planeof motionarerelatedby themoderadius. Themoderadiusis definedastheratioof

thetranslationandrotationmodeshapecoefficientsin thepredominantplaneof motion

with themodalvectorscaledto unit rotation. This simplifiedform of thesolution,given

by Equations3.32and3.33,will beusedto developa correctionfor vibration induced

errorsin Chapter4.

43

Chapter 4

MODEL ATTITUDE BIAS ERROR CORRECTION

4.1 Introduction

In this chapter, the theoretical model is used to develop the proposed time domain "modal

correction method" for model vibration induced errors in inertial wind tunnel model

attitude measurements. The modal correction theory and implementation procedure are

described. The proposed modal correction method extends the early work of Fuijkschot

[16] to compensate for multiple yaw and pitch vibration modes. This was the first time

domain correction technique developed to compensate for multiple modes of vibration in

the model pitch and yaw planes. A time domain correction is required due to the short

data acquisition periods (1 second) for the random wind tunnel data. This is also

important in order to meet future testing needs [21] involving the correlation of

instantaneous changes in model attitude and force balance data. The modal correction

method also minimizes the number of additional transducers required by using measured

modal properties of the wind tunnel model system. This is especially critical for models

with limited interior space and in wind tunnels that have extreme temperature conditions

where heated instrumentation packages are required.

Prior to the modal correction technique, the model attitude corrections were based on the

assumption that the instrumentation package moved on a circular arc with no detailed

analysis of the underlying system dynamics. The theoretical and experimental modal

44

analysesperformedduring thedevelopmentof themodalcorrectiontechniqueprovided

valuableinsight into thedynamicbehaviorof cantileveredwind tunnel modelsystems.

Observationof the relevantanimatedmodeshapesrevealedthat the modelmovedas a

rigid body on the more flexible sting-balancecombination. The assumptionof rigid

body modelmotion is critical to thedevelopmentof multi-modetime domaincorrection

techniques.

4.2 Modal Correction Theory

The primary generalized forces are associated with the "quasi-steady" aerodynamic loads

acting on the model. Unsteady flow in the wind tunnel results in a broadband random

input to the model system. The input for this process is not directly known or measured.

For the metallic sting-model structure, the damping is low and the system acts as a

narrow band filter passing energy (or responding) at the natural frequencies of the model

system [33]. If the modes are well separated and lightly damped, the response motion at a

natural frequency, _, will be described by the corresponding mode shape, {_ }r' with

residual effects of other modes assumed negligible.

45

The physicsof theproblemcannow be studiedby consideringthe responseof a single

modeasdepictedin Figure4.1. UsingEquation3.32,theresponsefor asingleyawmode

in simpleharmonicmotioncanbewritten

{_)(t))=

x

Y

Z

7

0

*y0

= 0

0

ry d_

" Pry (t)=f_fJry

ry

0

- Pry

0Pry0

0

1

sin(O3ry t) (4.1)

where Pry is a scalar constant related to the amplitude of motion. The reference

coordinates on the rigid fuselage will be taken at the location of the on-board inertial

angle of attack (AOA) package. The translation and rotation of the AOA package can

then be written

Yry (t) = Yry sin(O_ry t)(4.2)

1

_ry (t)= --Pry Yry (t) (4.3)

where Yry is a constant representing the amplitude of motion. Taking the derivative with

respect to time gives

Yry (t) = Vry cos(0lry t) where Vry = Yry Olry(4.4)

1

_ry (t)-- --Pry Yry (t) (4.5)

46

Yry

Modecenter of

Pry"

Figure 4.1 Harmonic motion of model at natural frequency of t.o_y.

47

The corresponding tangential and normal acceleration components, a t and a n , are:

at(t)= Yry (t)= Ary sin(COryt); where Ary =-VryO3ry (4.6)

an(t) = _ry (t)Yry (t)- --

Yry2(t)

-Pry

(4.7)

Substituting for J'ry from Equation 4.4 gives

Vry2

an(t ) ---z---_cos 2 (0_ry t)..... (1 + cos(2C0ry t))Pry 2 Pry

(4.8)

Recall from Chapter 1 that the on-board inertial AOA package uses a servo-accelerometer

with its sensitive axis parallel to the longitudinal axis of the model. The vibration

induced normal acceleration results in the AOA package sensing a centrifugal

acceleration coincident with its sensitive axis. The AOA package output prior to filtering,

Aunf , becomes:

Aun f (t) = g sin e_ + ff ry (t) _Yry (t) -- a x (t) (4.9)

The first term on the right hand side of the equation is the gravitational acceleration due

to the true model attitude, _, relative to the local vertical. The second term is the

centrifugal acceleration (from Equation 4.7) caused by the model yaw motion. The third

term represent the accelerations, ax(t) , resulting from flow induced longitudinal model

vibrations (typically greater than 50 Hz). In this equation, the positive output for the

AOA package corresponds to a positive change in angle of attack. Using the modal

48

radiusto relatethetranslationandrotationdegreesof freedom(Equation4.5) of the rigid

model,theequationcanbewritten

Yry2(t)Aunf (t) = g sins ax(t) (4.10)

Pry

Expanding Yry and using the trigonometric relations from Equation 4.8 gives

Aun f (t) = g sino_ - Vry2(l+cos(2_ryt)) -ax(t ) (4.11)

2pry

This form of the equation shows that the centrifugal acceleration for sinusoidal model

response results in the angle of attack sensor having a constant, bias, term and a harmonic

component at twice the natural frequency. The harmonic component and the longitudinal

acceleration, ax(t), can be removed by filtering. Lowpass filtering (0.4 Hz cut-off

frequency) the AOA signal yields

Vry 2Afil = g sinO_ (4.12)

2 Dry

The filtered AOA signal, Afil, has a bias error due to model vibration that cannot be

removed by filtering or averaging. From Equation 4.12, it is evident that in order to

remove the bias error, a correction method that compensates for both the amplitude of

vibration, Vry, and the mode shape, Pry, is required.

49

Model pitch vibrationcausesa similar biaserror term,wherethe tangentialvelocity is

actingin thepitchplane. If thevibrationresponseis composedof multipleyawandpitch

modes,thetotalbiaserrorwill bealinearsummationof theerrorcontributionsfor them

modes.

m Vr2Afi t = gsinot - _ (4.13)

r=l 2 Pr

Or, in terms of the peak acceleration, from Equation 4.6,

m Ar 2

Afil = gsino_ - Y_ 2 (4.14)r=l 20) r Pr

The above discussion is based on the case of continuous sinusoidal model motion. In the

wind tunnel, the data is random in nature. This results in a time varying bias error that is

dependent on the number of modes participating and the amplitudes of motion for those

modes. In order to compensate for a time varying bias errors, a time domain correction

appears to be the most suitable.

The proposed time domain modal correction technique is based on the single mode model

given by Equation 4.10. Assuming the model system behaves linearly, the total bias error

will be a linear superposition of the individual mode effects. This can be written as

Aunf(t)= gsin0_ - _ v2(t------2-ax(t)r=l Pr

(4.15)

50

whereVr(t) is the velocity (pitch or yaw plane) at the AOA location for mode r and Pr is

the corresponding mode radius. For m modes, the bias error estimate, aB(t), can be

written

aB(t)= _ v2(t)r=l Pr

(4.16)

Adding the bias error estimate to the unfiltered AOA output yields

Aunf (t)+ _ v2(t) N v2(t)- gsin0_ - ]_ ax(t)r=l Pr r=m+l Or

(4.17)

The longitudinal accelerations, ax(t), can be removed through low pass filtering. The

experimental data in Chapter 5 will show the majority of the dynamic response in the

pitch and yaw plane will be concentrated in the first four to six modes. Therefore, the

effects of the higher frequency modes (denoted by r=m+l to N) will be assumed

negligible. An estimate of the true model attitude is given by

O_(t) = sin -1

m v2(t)LP Aunf(t)+ Y_

r=l Pr

g(4.18)

where the accelerations are measured in g's and LPF designates a low pass filter with a

cut-off frequency of 0.4 Hertz.

In the modal correction technique, natural frequencies, f.or , and mode shapes, {_ }r must

first be determined. This can be done using analytical or experimental techniques. In

most cases, a detailed analytical model is not available. Experimental modal analysis

51

techniques[29, 30] havebeenusedto determinetherequirednaturalfrequencies,(Dr ,

and mode shapes, {t_ }r' of the cantilevered model systems. Recall from Chapter 3 that

the low frequency "rigid-model" modes of interest have predominant motion in the pitch

or yaw plane due to the model-balance design. This is shown graphically in Figures 4.2

and 4.3. For a given mode, the radius is estimated by assuming the fuselage moves as a

rigid body and using a least square linear regression fit of the fuselage mode shape

coefficients to determine an effective point of rotation (node). A vibration mode's

effective radius is estimated as the distance from the mode's point of rotation to the

inertial AOA sensor location in the model fuselage.

The rigid body assumption used in the mode radius estimation appears to be satisfactory

for the low frequency (<50 Hz) modes that are being evaluated. The accuracy of the rigid

body assumption can be assessed using the correlation coefficient for the linear regression

fit of the fuselage mode shape coefficients. For a linear regression fit of a yaw plane

mode (see Figure 4.2), the line estimate, Yi, is defined by

Yi = axi + b (4.19)

The correlation coefficient [34] is defined as

]_x_ yExy

n (4.20)

CC r = ]I(Y'X2- (_'x)21]_Y2(_'Y)2n n

52

Undeformed

x 5 x4Jr ÷

Y5 Y4

AOAposition

x2

Y2

Modecenter of

rotation--_ _ YT

Mode radius

P

Yi = a*xi + b

Figure 4.2 Yaw plane mode of model system.

53

AOAposition Mode radiusP

Zl 4....--I-

x 3 x2 Xlx 5 x4

Undeformedz i =a.x i + b

Mode / xcenter of/rotation -'

Figure 4.3 Pitch plane mode of model system.

54

The correlation coefficient is always between -1 and +1. For values close to zero, there is

no linear relationship. For values near _+1, there is a very strong linear relationship. In

Chapter 5, the correlation coefficient is used to assess the linear regression fit of the

measured fuselage mode shape coefficients.

A second assumption is that the mode shapes do not change significantly under the wind

tunnel test conditions. This enables wind-off estimates of the mode effective radii to be

used for correction of the model attitude measurement during wind tunnel testing. In

Appendix A, the effect of aerodynamic forces on the measured modal radius were

evaluated using a finite element model of a cantilevered wind tunnel model system. The

aerodynamic forces were applied to generate a prestressed model and then the

eigensolution was performed for this prestressed loading condition. For the largest

aerodynamic forces measured on a representative transport model in the National

Transonic Facility, the predicted shifts in the modal radius were less than 4%, which is

negligible.

4.3 Modal Correction Implementation

Once the effective radius and natural frequency are obtained for each mode of interest, the

next step in the modal correction technique is the on-line measurement of the unfiltered

AOA signal, and the lateral and normal accelerations at the AOA location. Due to the

model attitude accuracy requirements ( _+0.01 ° over a range of +20 ° ), a 16-bit analog-to

55

digital converteris requiredfor thedataacquisitionsystem.Oncethe datais acquired,

thedigitizedmeasurementsareprocessedoff-line usingMATLAB ®[35].

A flow chartof thedataanalysisroutineis shownin Figure4.4. Thelateralandnormal

accelerationmeasurementsarenumericallyintegratedusingthetrapezoidalrule [36] and

scaledto obtainthe lateraland normalvelocity, respectively. The velocity signalsare

squaredusing array,or elementby element,multiplication. For each lateralmode of

interest,a linearphasefinite impulseresponsefilter is usedto defineapassbandaboutthe

natural frequency. This isolatesthe velocity squaredcomponentsof the individual

modes. Thefilters areappliedin boththeforwardandreversedirectionsto obtainzero-

phasedistortionanddoublethefilter order. This is critical for a time domaincorrection

wherethe phaserelationshipof the unfilteredAOA signaland the lateral and normal

dynamicresponsemustbemaintained. Thesquaredvelocitycomponentsfor eachmode

are divided by their correspondingmode radius and then combined using linear

superpositionto give theestimatedbiaserrordueto lateraldynamics.This procedureis

then repeatedfor the normal,or pitch, modesto determinethe bias error due to pitch

dynamics. The errorsdue to the lateraland pitch dynamicsare thencombinedusing

linear superpositionto yield thetotal biaserror. Thebiasestimateis thenaddedto the

unfilteredAOA and the result is filtered with a 0.4 Hz lowpassfilter asdescribedby

Equation4.18. Thisgivesacorrectedtimevaryingmodelattitudesignalthatcanbeused

to determinetheinstantaneousor meanangleof attackoverthedataacquisitionperiod.

56

Start )

[ mp _pb_b i=l,nap

read accel, arraysay, a_, A..f

Vy_-integral(ay) ]Vz_-integral(az)

Vy2_.Vy.,VyVz2_" Vz'* Vz ]

()

apply ban@ass filter toisolate mode effect

Vyi2 (-BPFfy i( vy 2 )

1!

estimate ith mode bias [

mBy i_-Vyi 2/121yi I

Figure 4.4 Flowchart of modal correction method.

57

.%(".%y+.%,p

apply bandpass filter toisolate mode effect

Vpi2 4[-BPFfpi(Vp 2 )

estimate ith mode bias

mBpi_-Vp i2/Ppi

1AcC'A.n_Aa

&:fa4"LPFo.4Hz(&:)

AOA(- asi n(Acr a)* 180/pi

Figure 4.4(continued) Flowchart of modal correction method.

58

Chapter 5

EXPERIMENTAL VERIFICATION

5.1 Introduction

In this chapter, the modal correction method is verified through a combination of wind-

off dynamic tests on two transport model systems and wind tunnel test data. The modal

correction method is applied to wind-off model dynamic response data to compensate for

model vibration induced errors in the inertial model attitude measurement for defined

shaker inputs in the pitch and yaw plane. In addition, the modal correction method is

applied to measured dynamic response data recorded during wind tunnel testing of a high

speed transport model in the National Transonic Facility (NTF).

5.2 Wind-off Dynamic Response Tests

This section will describe the test setup and results of wind-off dynamic response tests on

two transport models [7, 8]. The modal correction method is validated for sinusoidal,

modulated sinusoidal and random inputs to the model in the pitch and yaw plane.

5.2.1 Test Setup and Procedure

Wind-off dynamic response tests were conducted on two transport models [7, 8] in a

model assembly bay at the National Transonic Facility. The test setup for the high speed

transport is shown in Figure 5.1. The mounting consists of a "rigidly" supported

cantilever sting that is positioned by a pitch-roll-translation mechanism. The model is

attached to the sting through a six component strain gage balance.

59

c_

c_

0E

°_,.q

E_

,_,.qLr_

The model was instrumented with an inertial AOA package [ 13] maintained at a constant

temperature of 160°F. The signal conditioner for the AOA package provides both an

unfiltered, "dynamic", 0 to 300 Hz bandwidth signal, and filtered, "static", 0 to 0.4 Hz

bandwidth signal. Two miniature accelerometers were installed on the face of the AOA

package to measure yaw and pitch motions. In addition, accelerometers were installed at

several locations on the model fuselage and sting to measure the dynamic response and

natural mode characteristics.

An experimental modal analysis was performed on the model systems. Frequency

response function data were acquired for point force excitation and transferred to a

personal computer. The STAR e [36] modal analysis software was used to determine the

modal parameters from the measured frequency response functions. A least square fit of

the fuselage mode shape coefficients was used to estimate the mode radius and

corresponding correlation coefficient (see Chapter 4).

For the dynamic response tests, an electrodynamic shaker was used to excite the model

system through a single point force linkage as shown in Figure 5.2. Due to the desired

high vibration amplitudes, the model surface was protected with tape and safety wire was

used in case the glue attaching the force mounting block failed during testing. The

excitation was applied in the pitch and yaw planes at the model fuselage hard points.

Sine, modulated sine and band limited random shaker input were used. A Hewlett

Packard (HP) 3566A dynamic signal analyzer was used to provide the shaker stimulus

and record the shaker force input, model force balance outputs, AOA static and dynamic

61

_iii!iiiii!iiii

¢)

r_O

¢)

e_

.,,._

outputs, and model accelerations. This system was used to monitor the model yaw and

pitch moments which established the dynamic test conditions for acquiring model attitude

measurements. Data was also recorded using a 16-bit Analog to Digital Converter (ADC)

board in a personal computer.

The model was set at a prescribed angle of attack under static conditions. The model

system natural frequencies were identified using sine sweep excitation in the pitch and

yaw planes. For each natural frequency of interest, a sinusoidal forced response test was

conducted by controlling the shaker input amplitude to provide a defined peak to peak

pitch or yaw moment on the model force balance. The control test variables were pitch

moment for modes that had predominantly pitch motion, and yaw moment for modes that

had predominantly yaw motion. The model attitude was measured at a series of moment

amplitude levels for sinusoidal excitation at a prescribed natural frequency of the model

system.

In addition to the sinusoidal forced response tests, the high speed transport model

dynamic response was measured for modulated sine and random excitation. The

modulated sine and random excitations and responses are more representative of the

model dynamics observed in actual wind tunnel tests. The majority of the modulated

sine tests were conducted with a 0.25 Hz modulation of the first natural frequency in the

pitch and yaw planes. In each case, the inertial AOA package was used to measure the

model attitude for a series of moment amplitude levels.

63

5.2.2 Commercial Transport Model Test Results

During wind tunnel tests, the commercial transport model had significant yaw vibrations

at 14 Hertz. Discrepancies in the aerodynamic data provided the stimulus for the

investigation of the AOA device [7] and its sensitivity to model vibrations. The AOA

investigation concentrated on the first four modes. An experimental modal analysis was

conducted on the model system and the results are tabulated in Table 5.1. Figures 5.3 and

5.4 show characteristic yaw plane modes described by sting bending and balance rotation.

The mode radii and corresponding correlation coefficients are also listed in the table.

Recall from Chapter 4 that correlation coefficients near +1 indicate a very strong linear

relationship. The correlation coefficient for the least square fit of the fuselage mode shape

coefficients shows the appropriateness of the linear regression fit and validates the rigid

body model assumption for the tabulated modes.

Table 5.1

Modal Parameters for Commercial Transport Model

Mode Frequency Damping Radius Corr.

No. (Hz) (%) (Inch) Coeff.

1 10.3 1.01 38.2 .9998

2 11.2 1.78 70.5 .9971

3

4

14.4

16.5

0.46

0.59

7.05

12.0

-.9973

.9998

Mode Description

Sting Bending-Yaw Plane

Sting Bending-Pitch Plane

Model Yaw on Balance

Model Pitch on Balance

64

"0

_ E

_r r

0

0°l,-q

t_

c5

° ,,..q

_0

e_

_r_

_0°_,.q

LT_

\r

0

0.,-_

I-i

°,,=wp,

d

,.Q

0

b_

0

N

.,=,_

LI.

For the AOA investigation, the model system was locked at near zero degree angle of

attack under static conditions. Single frequency forced response tests were conducted by

controlling the shaker input to provide a defined peak to peak pitch or yaw moment on

the model force balance. The test variable was yaw moment for modes that had

predominantly yaw motion, and pitch moment for modes that had predominantly pitch

motion. The AOA response data and model accelerations were recorded for several

moment levels. This data was transferred to the MATLAB ® [35] program for application

of the modal correction technique. The measured mean AOA output, estimated bias, and

corrected mean AOA output, after application of the modal correction technique, are

shown versus balance moment in Figures 5.5 through 5.8. Recall that for sinusoidal

input, the model vibration creates a bias error or offset in the mean value. After

application of the modal correction technique, the error is reduced to the AOA device

accuracy of +0.01 degrees for all measurements except the second pitch mode. For this

case, an order of magnitude reduction is obtained.

The accuracy of the correction for the pitch axis tests may be improved by locating the

accelerometers adjacent to or inside the heated AOA package. The pitch plane

accelerometer on the face of the AOA package failed early in the test. A triax set of

accelerometers located externally on the fuselage upper surface was subsequently used to

obtain the off-axis accelerations required for the modal correction technique.

67

0.02

0

"_m -0.02

g-0.04

i -0.06

-0.08

-0.1

400

_""" _- ... ......... • .............. &....-" .... .--''_

_...__..-__._..._:-_-...............................

-- • -- Estimated Bias "X_\

- - _r -- Corrected "X_\

° \I I I

800 1200 1600 2000

Yaw Moment (Inch-Pounds)

Figure 5.5 Measured mean AOA, estimated bias, and corrected mean AOA versus

yaw moment for sinusoidal input at 10.3 Hz.

68

A

D1Oo

u

,¢

O

e

t,-,¢

0.02

-0.02

-0.04

-0.06

-0.12

-0.14

._.. . . ...... dr .......... • ........... • ............................................

I'" _''" Corrected "_\

I .... N°minal +'01° "_

1.... Nominal-.01 ° _

-0.16 , ' '

1000 4000 6000 8000 10000 12000




69

0.02

-0.04

400

m

I-...... ...-A ........... • ......,-... ,.: :, - - _- .......... • .....

"B.......... .'_,.=,,. ............................. _.._

----e--- Measured

- •- Estimated Bias

- - _" • Corrected

.... Nominal +.01 o

.... Nominal-.01 °J I I I

800 1200 1600 2000 2400

Pitch Moment (Inch-Pounds)


pitch moment for sinusoidal input at 11.2 Hz.

70

0.05

-0.05

"_ -0.1

-o.15

i -0.2-0.25

-0.3

-0.35

-0.4

'-"--';:-'- ........-'- "-"• ,r-'-'---.:---'.'.-.'--::-::-_.'.--;-.-"o

Measured _" ,,

- •- Estimated Bias -,,'_"11

- - _- - "Corrected "_ \

I I I I

4000 6000 8000 10000 12000 14000




71

5.2.3 High Speed Transport Model Test Results

A high speed transport model system that experienced high levels of vibration [6] during

previous wind tunnel tests was selected to further investigate the effects of dynamics on

the inertial AOA package. Measurements taken during wind tunnel tests indicated that

the primary modes being excited were at approximately 8-10 Hz and 28-30 Hz [6]. An

experimental modal analysis of the model system was conducted and the results are listed

in Table 5.2. The radii and corresponding correlation coefficients for the vibration modes

were estimated using a least square linear regression fit of the modal deformations as

described in Chapter 4. The correlation coefficient for the least square fit of the fuselage

mode shape coefficients shows the appropriateness of the linear regression fit and

validates the rigid body model assumption for the tabulated modes.

Table 5.2

Modal Parameters of High Speed Transport Model

Mode

No.

Frequency

(Hz)

Damping

(%)

Radius

(Inch)

6

9.0

9.2

20.5

21.7

29.8

34.9

1.32

1.68

2.75

2.70

2.28

2.59

31.0

30.2

0.18

-1.07

-7.16

-7.65

Corr.

Coeff.

.9997

.9997

-.9993

.9999

-.9983

0.9999

Mode Description






with Sting Second Bending


with Sting Second Bending

72

It is important to note that the mode radius may be positive or negative dependent on the

vibration mode shape. Previously, this bias error was described as a "sting whip" [13]

error and associated with the first sting bending modes in the pitch and yaw planes. The

analyses and experimental data presented in this dissertation show that the model system

dynamics is more complex than previously assumed. The physical interpretation of the

sign of the radius is more easily understood by examining the 9.0 Hz and 29.8 Hz yaw

modes shown in Figures 5.9 and 5.10. For the case where the radius is negative, the point

of rotation for the vibration mode is forward of the AOA package. A positive radius is

defined for a point of rotation aft of the AOA package.

The significance of the sign of the radii is that the bias error may be positive or negative

dependent upon the vibration mode being excited. This is demonstrated by the response

of the two yaw plane modes shown in Figures 5.11 and 5.12. For the 9.0 Hz yaw mode,

the indicated model angle change is negative when the model is being driven with

sinusoidal excitation at the natural frequency and then returns to its nominal angle when

the shaker system is shutoff. The 29.8 Hz yaw mode, which has a negative radius value,

shows an indicated positive angle change when the model is being driven with sinusoidal

excitation at the natural frequency and then returns to its nominal angle when the shaker

system is shutoff. The excitation system was adequate to show the above trends,

however, only the first mode in each the yaw and pitch planes were excited to levels that

showed significant shifts in the indicated model attitude from the onboard inertial AOA

package. Difficulty in driving the higher frequency modes is attributed to the rigid

73

r

0

0. i,-.q

L.

. ....q

N=o.

.8

° ,,....q

r_

(D

E_o E

/l:nr-

,D

09

I

I

I

I

>,

0

o

,.o°,.,_

N

t",l

0

0

0

Inertial AOA

4.34 .............................................. ............................................ .......................................................................................................................................

Deg I

4.14i i : .......... !

0 Sec 8 Sec

Yaw Acceleration

1.5 .......................... !_........................ ............ ......

g

-1.5.....................................i..................................................................................i.....................................i.............................................0 Sec 8 Sec

Yaw Moment

3600 ............................................. ............................................ ...................... ...............................................................................................................

In-Lbs

-2400 _: ................. i.............................0 Sec 8 Sec

Figure 5.11 Inertial AOA measurement, yaw acceleration, and yaw moment versus

time for 9.0 Hz sinusoidal input in yaw plane.

76

0.4g

-0.40 Sec 8 Sec

Yaw Moment

2400 ....................i...........................................:................................................................................................................................................ii

In-Lbs

............ ].......................

-600

0 Sec 8 Sec

Figure 5.12 Inertial AOA measurement, yaw acceleration, and yaw moment versus

time for 29.8 Hz sinusoidal input in yaw plane.

77

backstop support in the model assembly bay. During previous wind tunnel tests [6], the

model coupled with the model support structure resulting in high dynamic yaw moments

with energy in the 28-30 Hz band. This points out the need to do dynamic testing with

the model installed in the tunnel.

The results of sinusoidal excitation tests for the first mode in each the yaw and pitch

plane are shown in Figures 5.13 and 5.14. The model was set at a nominal angle of 0 ° for

these tests. For a set excitation level, time domain data were acquired and stored using

the dynamic signal analyzer. These data were transferred to a personal computer where

the modal correction technique, implemented in an m-file in the MATLAB ® [35]

language, was used to estimate the bias error in the inertial device. This procedure was

repeated for several excitation levels as defined by the moment amplitude level.

As shown in Figures 5.13 and 5.14, the estimated bias error is in good agreement with the

indicated mean angle change measured with the onboard inertial AOA sensor. After

application of the modal correction method, the bias error is reduced from a maximum of

-0.146 ° to -0.009 ° for the first mode in the yaw plane and from -0.175 ° to-0.006 °

for the first mode in the pitch plane. These corrected mean angle of attack values are

within the AOA accuracy requirement of 0.01 °. Similar results were obtained for

sinusoidal input tests with the model set to nominal angles of 4.3 ° and 6 °.

78

0.02

A

t_

0

o

0

_.eo}e-

-0.02

-0.04

-0.06

-0.08

-0.1

-0.12

-0.14

-0.16

9OO

__S._ _-_*.__;_L; -::-._.--._:: _:: ._':----;--.-...

- • - Estimated Bias "X_\

oreC;%o.... Nominal -.01 o

I i I

1800 2700 3600 4500

Yew Moment (Inch-Pounds)



79

0.02

A

w

2o

um

0

e

i-<

-0.02

-0.04

-0.06

-0.08

-0.1

-0.12

-0.18

1000

m

.......... ,- -- : ;:.,:._.:: ._.::_,_-,_ .........

Measu re_ _ \ \

-O--_--M_amSU:: Bias "_

- - _- - • Corrected

I I, I

2000 3000 4000


50O0



80

In order to obtain a corrected time domain angle of attack measurement that can be used

for instantaneous or average values, it is important to maintain the phase relationship

between the measured and estimated bias error. To verify that the modal correction

method maintains this phase relationship, the bias error was examined for modulated

sine and random inputs. The measured response for modulated sine and random inputs is

also more representative of actual wind tunnel test data.

Figure 5.15 shows the measured angle of attack and estimated bias error as a function of

time for a 9.2 Hz pitch excitation with a 0.25 Hz modulation. Excellent agreement is

obtained with the difference between the measured angle of attack and estimated bias

error being less than 0.005 ° . Modulated sine tests were conducted at several excitation

amplitude levels for the first mode in each the y and z axes and consistent results were

obtained between the measured angle of attack and predicted bias errors for all cases.

In addition, the response of the AOA package for two levels of random excitation in the

pitch plane were also examined. Figure 5.16 shows an eight second record of the inertial

AOA sensor response for the highest level random excitation. The random response

measured by the pitch accelerometer on the face of the AOA package was composed of

primarily 9.2 Hz response. The bias error estimate based on only the 9.2 Hz mode

contribution is also shown in Figure 5.16. Again, the measured angle of attack and

estimated bias error are in very good agreement.

81

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5.3 High Speed Transport Model Wind Tunnel Tests

Dynamic response studies were conducted on a high speed transport model installed in

the test section of the NTF. The dynamic response characteristics were also recorded for

high speed (Mach=0.95) wind tunnel runs.

5.3.1 Test Setup in Wind Tunnel

The model was instrumented with a re-designed inertial AOA package that has two servo-

accelerometers for measuring model AOA and two dynamic accelerometers to measure

the accelerations tangent to the sensitive axis of the AOA sensors. The package is

maintained at a constant temperature of 160°F. The signal conditioner for the AOA

sensors provide both an unfiltered, dynamic, 0 to 300 Hz bandwidth signal and a filtered,

static, 0 to 0.4 Hz bandwidth signal.

Initial wind-off dynamic response studies were performed in the wind tunnel test section

using shaker excitation of the model with the arc sector in a fixed position. For the wind-

off shaker excitation tests, six additional accelerometers were mounted external to the

model fuselage to measure model yaw and pitch motion at three locations.

Data were acquired using a 16 channel digital data acquisition system with 16-bit

resolution. All dynamic signals were filtered to 100 Hz prior to recording. Data were

recorded at 200 samples per second per channel. Recorded channels included the dynamic

and static inertial AOA outputs, the tangential accelerations in yaw and pitch, and the six

84

forcebalancecomponents.Datawererecordedfor both the wind-off shakerexcitation

testsandthehighspeedwind tunnelruns.

Forthewind-off shakerexcitationtests,a HewlettPackardmodel3566Adynamicsignal

analyzerwasusedto providetheshakerstimulusandperformon-linetime andfrequency

domainsignalanalysis.The 16channelsignalanalyzerwasusedto monitorand record

theshakerforceinput,andtheresponseof thesix accelerometersmountedexternalto the

modelfuselage.

The shakerexcitationtestswereperformedwith themodel installedin the test section

andthearc sectorin a fixed position.An electrodynamicshakerwasusedto excitethe

model in theyaw planethrougha singlepoint force linkage13 inchesaft of the model

nose. Due to scheduleconstraints,theforcedresponsetestswereconductedin the yaw

plane only. The model systemnatural frequencieswere identified using sine sweep

excitation. ThedynamicandstaticinertialAOA outputs,thetangentialaccelerationsin

yawandpitch,andthesix forcebalancecomponentswererecordedfor a seriesof shaker

forceamplitudelevels for sinusoidalexcitationat a prescribednaturalfrequencyof the

model system. In addition to the sinusoidalforced responsetests, modulatedsine

excitationtestswereperformedfor a seriesof shakerforce levels. The modulatedsine

excitationsand responsesaremore representativeof the model dynamicsobservedin

actualwind tunneltests.

85

For agiventestcondition,timedomaindatawereacquiredandstoredon the 16-channel

dataacquisitionsystem.Thesedataweretransferredto apersonalcomputerwherea

softwareroutineimplementingthemodalcorrectionmethod,writtenasanM-file in the

MATLAB ®[35] language,wasusedto estimateandcorrectfor thebiaserror in the

inertial device.

5.3.2 Dynamic Response Tests in Wind Tunnel

An experimental modal analysis was performed for a high speed research model installed

in the NTF wind tunnel and the dominant modes are listed in Table 5.3. The model was

configured differently than in previous wind-off vibration tests, therefore, the modal

characteristics are different than those presented in the previous section. The mode radii

and corresponding correlation coefficients are also listed in the table. The correlation

coefficients again confirms the rigid body model assumption.

Table 5.3

Modal Parameters for Survey of High Speed Transport Model in Test Section

Mode

No.

1

2

3

4

5

6

Frequency

(Hz)

7.3

9.8

12.1

16.9

17.2

21.1

Damping

(%)

0.46

0.28

0.51

1.3

1.0

0.36

Radius

(Inch)

37.8

31.8

8.71

-0.93

-3.40

-9.54

Corr.

Coeff.

.9992

.9995

.9995

-.9985

-.9998

-.9994

Mode Description



Sting/Model Yaw



Model Yaw, 2nd Sting

Bending

86

Theresultsof sinusoidalexcitationtestsfor thefirst mode(7.3Hz) in theyawplaneare

shownin Figure5.17. This figure showstheangleof attackmeasuredwith theprimary

servo-accelerometersensorand the correctedangle of attack after removal of the

dynamicallyinducedbiaserror. Thesetestswereconductedwith themodelat a nominal

angleof 6.01° and the arc sectorin a fixed position. After applicationof the modal

correctionmethod,the error is reducedfrom a maximumof -0.087° to +0.003° for the

first modein theyawplane. As shownin Figure5.17,thecorrectedAOA measurements

are within the AOA accuracy requirement of +/- 0.01 o. The higher frequency modes were

not excited to high enough levels to produce significant shifts in the AOA measurements

during the wind-off vibration tests.

In addition to the sinusoidal tests, the bias error was examined for modulated sine input.

Figure 5.18 shows the angle of attack measured with the primary servo-accelerometer

sensor and the corrected angle of attack after removal of the dynamically induced bias

error. This data was obtained for excitation at the 7.3 Hz natural frequency with a 0.5 Hz

modulation. The corresponding measured yaw moment is also shown in Figure 5.18 and

has a maximum peak-to-peak value of 2400 in-lbs. Excellent correction is obtained

using the modal correction method with errors as large as -0.091° being reduced to less

than +/- 0.005 ° from the nominal angle. Modulated sine tests were conducted for the first

mode at several excitation amplitude levels and consistent results were obtained. For this

type of model response, correction for the dynamically induced errors results in a shift in

the mean value and a reduction in the variance of the signal.

87

6.02

,01 .............................................. " ......

6

_ 5.99

5.98

5.97

o 5.960

< 5.95

5.94

5.93

5.92

0 400 800 1200 1600 2000 2400 2800


Figure 5. I7. Measured and corrected angle-of-attack for sinusoidal excitation at 7.3Hz.

88

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5.3.3 Wind Tunnel Test Results

The data for the first 64 seconds of a test on the high speed transport model (Mach=0.95,

Q=1800 pounds-per-square-foot, T=-254°F) were used to evaluate the dynamic response

characteristics of the primary AOA sensor and the proposed modal correction method.

The filtered output of the primary AOA is shown in Figure 5.19. Data analysis was

restricted to the periods where the model pitch angle was paused to obtain "steady-state"

aerodynamic data. The pitch acceleration was significantly lower than the yaw

acceleration over the data analysis period. Analysis of the model yaw acceleration

showed primarily 7.3 Hz response with additional energy at the 12.1 Hz natural

frequency. Intermittent response at other frequencies was observed. Initial application of

the modal correction method included the modes in Table 5.3. The AOA mean value and

standard deviation over each pause period are listed in Table 5.4.

Time Period

(Seconds)

Table 5.4

Summary of Wind Tunnel Results

Measured

AOA Mean

(Degrees)

Measured AOA

Standard

Deviation

(Degrees)

Corrected

AOA Mean

(Degrees)

-3.5403

Corrected AOA

Standard

Deviation

(Degrees)

0.01210to 9.25 -3.5664 0.0179

12.5 to 32.5 -2.5094 0.0203 -2.4764 0.0082

40.75 to 52.5 -1.4803 0.0248 -1.4392 0.0088

56to 64 -0.9308 0.0094 -0.9121 0.0057

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with the primary servo-accelerometer sensor and the corrected angle of attack after

removal of the dynamically induced bias error for each pause period. There were no

optical measurements to confirm the corrected AOA measurements.

Since the response was primarily at the 7.3 Hz and 12.1 Hz natural frequencies, which

have positive radii, trends consistent with the wind-off modulated sine test are expected.

For a mode with a positive radius and fluctuating amplitude of motion, correction for

dynamically induced errors will result in a positive shift in the mean value and a reduced

variance for the corrected signal. The significant reduction in the variation observed in

the corrected time domain AOA signal (Figures 5.20-5.23) as compared to the measured

primary AOA and the corresponding reduction in the standard deviation for the corrected

AOA measurement indicate successful application of the modal correction method. The

periods from 12.5 to 32.5 seconds and 40.75 to 52.5 seconds (part of which are shown in

Figures 5.21 and 5.22) are the best indicators of the amount of bias reduction possible.

The inclusion of more natural frequencies in the modal correction method may aid in

improving the bias correction. It is also important to note that the low frequency

fluctuations in the corrected AOA signal may be due in part to oscillatory changes in the

model pitch attitude.

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For this test, the data acquisition periods were longer than normal. The steady state data

at NTF is typically taken over a 1 second period. Differences between the corrected and

measured mean values over a given one second interval may be much larger than those

shown in Table 5.4. The results for one second intervals from 16 to 22 seconds are listed

in Table 5.5. Differences between the measured and corrected AOA mean value as large

as -.064 ° are observed over the selected one second intervals.

Table 5.5

Summary of Wind Tunnel Results for One Second Data Acquisition Periods

Time Period

(Seconds)

Measured AOA

Mean

(Degrees)

Corrected AOA

Mean

(Degrees)

Difference

Measured -Corrected

(Degrees)

16 to 17 -2.531 -2.473 -0.058

17 to 18 -2.513 -2.483 -0.030

18 to 19 -2.508 -2.478 -0.030

19 to 20 -2.550 -2.486 -0.064

20 to 21 -2.540 -2.481 -0.059

21 to 22 -2.511 -2.487 -0.024

97

Chapter 6

CONCLUDING REMARKS

An original system dynamic analysis approach is presented to evaluate the effects of

model vibrations on measured aerodynamic wind tunnel data. Analytical and

experimental results show that centrifugal accelerations associated with model vibration

cause bias errors in the inertial model attitude measurements. Wind-off dynamic response

tests on two transport model systems found bias errors over an order of magnitude greater

than the required device accuracy. An analysis is presented that shows these errors can

not be removed by filtering or averaging. Equations are developed to show the influence

of the model attitude errors on the determination of the drag coefficient.

A new time domain technique is developed to correct for the dynamically induced errors

in the inertial model attitude measurements using measured modal properties of the

model system. This modal technique extends previous work to compensate for multiple

modes in the pitch and yaw plane. Previously, the problem was associated with "sting

whip" with no detailed analysis of the underlying system dynamics. Dynamic response

tests on two transport models in a laboratory environment demonstrated the need to

compensate for multiple modes. Theoretical and experimental modal analyses are

presented to provide physical insight into the model system dynamics. Based on

observed rigid body model motion for the low frequency modes of interest, the problem is

simplified. For a planar rigid body model mode, analysis shows that the fuselage motion

can be completely described by a translation and rotation degree of freedom. A mode

98

radiusis definedto relatethetranslationandrotationdegreesof freedomusinganalytical

or experimentalmodeshapes.Analyses are presented that show the mode radii are not

affected significantly by the aerodynamic loads experienced in a high dynamic pressure

wind tunnel environment. A correlation coefficient is defined and used to validate the

rigid body model assumption.

Due to short data acquisition periods and the multi-mode random response observed in

wind tunnels, state of the art digital signal processing techniques are required to

implement the modal correction method in the time domain. Bandpass filters are used to

isolate the effects of individual modes and then the mode effects are combined using the

principle of superposition. During the filtering processes, the phase relationship of the

unfiltered model attitude signal and the model dynamic response must be maintained. To

achieve zero-phase distortion, finite impulse response filters are applied in both the

forward and reverse directions. The modal correction method compensates for the

dynamically induced bias error and provides a corrected model attitude time signal that

can be used to correlate with time varying changes in the balance forces

The modal correction method is verified through a series of wind-off dynamic response

tests and actual wind tunnel test data. The wind-off dynamic response tests show the

method has the ability to reduce the bias error in the inertial model attitude device by over

an order of magnitude to achieve the required device accuracy.

99

Theoreticalandexperimentalresultsarepresentedthat demonstratethe needto correct

for dynamicallyinducederrorsin inertialwind tunnelmodelattitudemeasurements.A

correctionmethodrequiringfour additionaltransducerswasdevelopedandimplemented

at the NationalAerospaceLaboratoryin the Netherlands.A principal advantageof the

modalcorrectiontechniqueis thatit minimizesthenumberof requiredtransducers(two)

usingthe modalpropertiesof the modelsystem. This is especiallycritical for models

with limited interior spaceandin wind tunnelsthathaveextremetemperatureconditions

where heated instrumentation packages are required. Recently redesigned

instrumentationpackagesfor the National TransonicFacility (NTF) provide the two

additionaltransducersrequiredfor themodalcorrectionmethod. Currently,facilities in

theUnitedStateshavenotimplementedacorrection.

Futureresearchof wind tunnel model systemdynamicsand its effectson measured

aerodynamicdata is recommendedin the following areas: (1) Perform a statistical

analysisto evaluatethe significanceof the magnitudeof the angleof attackcorrection

with respectto the measuredstandarddeviation,and small angleassumptionfor high

anglesof attack; (2) Performastudyof thecrossaxissensitivityof the inertial attitude

sensor,andtheeffectsof modelroll motions; (3) Performa studyof alternatesignal

processingmethods,such as modulationtechniques,for removing the dynamically

inducederrorsin the inertialmodelattitudemeasurements;(4) Basedon theobserved

rigid body modelbehavior,performa parametricstudyto evaluatechangesin dynamic

responsefor variationsin: massor massdistributionof themodel;balancestiffnessand

100

damping;and sting materialproperties. This researchwould be aimedat developing

designcriteriafor modelsystemsthatwouldminimize themodeldynamicresponseand

move closer to the desired steady-statewind tunnel test conditions. Further

enhancementsmaybe foundin theuseof activevibrationcontrol techniquesto suppress

themodelvibrations.

101

[1]

[2]

[31

[41

[51

[6]

[7]

[81

[9]

[10]

[11]

Chapter 7

REFERENCES

Young, C. P., Jr.:"Model Dynamics", AGARD Special Course on Cryogenic

Wind Tunnels, 1996.

Fuller, D.E.: "Guide to Users of the National Transonic Facility",

NASA TM-83124, July, 1981.

Strganac, T. W.: "A Study of the Aeroelastic Stability for the Model Support

System of the National Transonic Facility", AIAA-88-2033, 1988.

Whitlow, W., Jr.; Bennet, R. M.; and Strganac, T. W.: "Analysis of Vibrations of

the National Transonic Facility Model Support System Using a 3-D Aeroelastic

Code", AIAA-89-2207, 1989.

Young, C. P., Jr.; Popernack, T. G., Jr.; Gloss, B.B.: "National Transonic Facility

Model and Model Support Vibration Problems", AIAA-90-1416, 1990.

Buehrle, R. D.; Young, C. P., Jr.; Balakrishna, S.; and Kilgore, W. A.:

"Experimental Study of Dynamic Interaction Between Model Support Structure

and a High Speed Research Model in the National Transonic Facility",

AIAA-94-1623, 1994.

Young, C. P., Jr.; Buehrle, R. D.; Balakrishna, S.; and Kilgore, W.A.: "Effects ofVibration on Inertial Wind-Tunnel Model Attitude Measurement Devices".

NASA Technical Memorandum 109083, August, 1994.

Buehrle, R. D.; Young, C. P., Jr.; Burner, A. W.; Tripp, J. S.; Tcheng, P.; Finley,

T. D.; and Popernack, T. G., Jr.: "Dynamic Response Tests of Inertial and OpticalWind-Tunnel Model Attitude Measurement Devices", NASA Technical

Memorandum 109182, February, 1995.

Pope, A.; and Goin, K. L.: High Speed Wind Tunnel Testing, John Wiley &

Sons, Inc., New York, 1965.

Muhlstein,L. ,Jr.; and Coe, C. F.: "Integration Time Required to Extract Accurate

Data from Transonic Wind-Tunnel Tests", Journal of Aircraft, Volume 16, No. 9,

pp 620-625, September 1979.

Mabey, D. G.: "Flow Unsteadiness and Model Vibration in Wind Tunnels at

Subsonic and Transonic Speeds", Royal Aircraft Establishment Technical Report

70184, October, 1970.

102

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

Steinle, F. and Stanewsky, E.: "Wind Tunnel Flow Quality and Data Accuracy

Requirements", Advisory Group for Aerospace Research and Development

(AGARD) Advisory Report No. 184, November, 1982.

Finley, T., and Tcheng, P.: "Model Attitude Measurements at NASA Langley

Research Center", AIAA-92-0763, 1992.

Burt, G. E., and Uselton, J. C.: "Effect of Sting Oscillations on the Measurement

of Dynamic Stability Derivatives in Pitch and Yaw", AIAA Paper No. 74-612,

July, 1974.

Billingsley, J. P.: "Sting Dynamics of Wind Tunnel Models", Arnold Engineering

Development Center Report Number: AEDC-TR-76-41, May, 1976.

Fuijkschot, P. H.: "Use of Servo-Accelerometers for the Measurement of

Incidence of Windtunnel Models", National Aerospace Laboratory, The

Netherlands, Memorandum AW-84-008, 1984.

Buehrle, R. D.; and Young, C. P., Jr.; "Modal Correction Method for

Dynamically Induced Errors in Wind-Tunnel Model Attitude Measurements",

Proceedings of the 13th International Modal Analysis Conference, pp. 1708-1714,

Nashville, Tennessee, February 13-16, 1995.

Tcheng, P.; Tripp, J. S.; and Finley, T. D.; Effects of Yaw and Pitch Motion onModel Attitude Measurements, NASA Technical Memorandum 4641, February

1995.

Fuijkschot, P. H.: "A Correction Technique for Gravity Sensing Inclinometers",

National Aerospace Laboratory, The Netherlands, Memorandum AF-95-004, 1995

Fuijkschot, P. H.: "A Correction Technique for Gravity Sensing Inclinometers-

Phase 2: Proof of Concept", National Aerospace Laboratory, The Netherlands,

CR 95458L, 1995.

Gloss, Blair, B.; "Future Experimental Needs To Support Applied Aerodynamics:

A Transonic Perspective", AIAA Paper 92-0156, 1992.

Owen, F. K.; Orngard, G. M.; McDevitt, T. K.; and Ambur, T. A.; "A Dynamic

Optical Model Attitude Measurement System", European Transonic Windtunnel

GmbH and DFVLR, Cryogenic Technology Meeting, 2nd, Cologne, West

Germany, June 28-30, 1988, Paper, 21 p.

Roberson, J. A.; and Crowe, C. T.: Engineering Fluid Mechanics, Houghton

Mifflin Company, 1980.

103

[35]

[36]

[37]

[38]

[24] Meirovitch, Leonard: Elements of Vibration Analysis, McGraw-Hill, Inc., 1975,

p 240-250.

[25] Wells, D. A.: Schaum's Outline of Theory and Problems of Lagrangian

Dynamics, Schaum Publishing Company, 1967.

[26] Tse, F. S.; Morse, I. E.; and Hinkle, R. T.: Mechanical Vibrations Theory and

Applications, Allyn and Bacon, Inc., 1978.

[27] Fung, Y. C.: An Introduction to the Theory of Aeroelasticity,

John Wiley & Sons, Inc., 1955.

[28] Craig, R. R., Jr.: Structural Dynamics: An Introduction to Computer Methods,

John Wiley & Sons, Inc., 1981.

[29] Allemang, R. J.; and Brown, D.L.: Chapter 21: Experimental Modal Analysis,Shock and Vibration Handbook, 3rd Edition, McGraw Hill, Inc., 1988.

[30] Ewins, D. J.: Modal Testing: Theory and Practice, Research Studies Press LTD.,

1984

[31] Ferris, A.T.: "Cryogenic Wind Tunnel Force Instrumentation", NASA

Conference Publication No. 2122, Part II, 1982, pp 299-315.

[32] Thomson, W. T.: Vibration Theory and Applications, Prentice- Hall, Inc., 1965,

pp. 179-182.

[33] Davenport, A. G.; and Novak, M.: Chapter 29 Part II: Vibration of Structures

Induced by Wind, Shock and Vibration Handbook, 3rd Edition, McGraw Hill,

Inc., 1988.

[34] Alder, H. L.; and Roessler, E. B.: Introduction to Probability and Statistics,

W. H. Freeman and Company, 1960.

MATLAB Reference Guide, The Math Works Inc., August, 1992.

Hornbeck, R. W.; Numerical Methods, Quantum Publishers, Inc., 1975.

The STAR System User Manual, Spectral Dynamics, Inc., 1996.

MSC/NASTRAN User's Manual, The MacNeal-Schwendler Corporation, 1989

104

Appendix A

EFFECT OF AERODYNAMIC FORCES ON MODAL CHARACTERISTICS

Introduction

In this section, the effect of aerodynamic forces on the modal characteristics of a

cantilevered wind tunnel model system are examined. The objective is to validate the

assumption that the modal characteristics do not change significantly under the wind

tunnel test conditions. This is a fundamental assumption of the modal correction method

that enables wind-off estimates of the natural frequencies and mode effective radii to be

used for correction of the model attitude measurement during wind tunnel testing. A

finite element model (FEM) of a representative cantilevered transport model is used as

the basis for evaluating the modal characteristics for several loading conditions including

the most severe forces measured in a recent wind tunnel test on this model in the National

Transonic Facility (NTF).

Analytical Model

The finite element model of a representative cantilevered transport model system for the

NTF was generated and analyzed using the MSC/NASTRAN ® [38] structural analysis

program. The FEM was developed with the goal of representing the low frequency (less

than 50 Hertz) "rigid-fuselage" modes that contribute to the errors in the inertial model

attitude measurements. Detailed modeling of the wings was not of interest for this study.

The sting and model fuselage are constructed of beam elements with equivalent material

105

and cross-sectionspecificgeometricproperties. The forcebalancewhich connectsthe

stingto the fuselagewasmodeledusinga concentratedmassequalto thebalancemass

and rigid bar elements. Springswere usedat the connectionbetweenthe rigid bar

elementand the fuselageto representthe balancestiffnesscorrespondingto the three

translationandthreerotationdegreesof freedom.Thebalancestiffnesswasdetermined

from experimentalmeasurements.The wings are modeled as concentratedmasses

attachedto thefuselageusingrigid barelements. An additionallumpedmasswasused

to representinstrumentationandassociatedhardware.

The primary generalizedforcesare the unsteadyaerodynamicloads.

loads are modeled using a quasi-steadyapproximation [27].

aerodynamicforcesaremodeledas:

QF = q_ × S × CF

The aerodynamic

The generalized

(A.I)

where, q_ is the dynamic pressure and S is the characteristic area. The coefficient CF will

be assumed linear and is a function of the model attitude. Similarly, the generalized

aerodynamic moments are modeled as:

QM = qoo x S × d × C M (A.2)

where d is the characteristic length and the coefficient CM is assumed linear and is a

function of the model attitude.

Data from a high-speed (Mach =0.9, q==1800 pounds per square foot) wind tunnel test of

this transport model in the NTF were used to determine the four most severe loading

106

conditions. To add additional conservatism,this data was scaledup to a dynamic

pressureof 2700poundsper squarefoot. The resultingloadingconditionsare listed in

Table 1. Theseforcesandmomentswereappliedto theFEM at a point on the fuselage

coincidentwith thebalancemomentcenter.

Table 1Transport Model

Worst Case Loading Conditions

Load Case

1

Axial Force

(Pounds)

69

Normal Force

(Pounds)-2271

Pitch Moment

(Inch-Pounds)4OOO

2 63 -491 3158

3 -53 2688 1474

4 -184 6035 632

For each of the four different aerodynamic load cases, a static analysis was run to

generate a prestressed model and then the eigensolution was run for this prestressed

loading condition. The eigensolution was also run for the no load case to provide a

baseline set of natural frequencies and mode shapes.

Results and Conclusions

The purpose of the analysis was to assess the effect of aerodynamic loading on the modal

characteristics of a cantilevered wind tunnel model system. For the research presented in

this dissertation, an important constant is the modal radius which is estimated from a

linear regression fit of the fuselage mode shape coefficients. Therefore, the comparison

107

criteriaarenaturalfrequenciesandmoderadii.Themoderadiusfor thefirst six analytical

modeswereestimatedusingthemethoddescribedin Chapter4. The naturalfrequencies

and mode radii for the different loading conditions are listed in Tables 2 and 3,

respectively. The naturalfrequencydoesnot shift significantlyfor anyof the loading

conditions.For the largestaerodynamicforcesmeasuredon a representativetransport

model in the NationalTransonicFacility, the predictedshifts in the modal radiuswere

lessthan4%,which is negligible.

Table 2

Transport Model

Natural Frequency Comparison

No Load Load

Case 1

Mode Frequency Frequency

(Hz) (Hz)

1 9.19 9.19

2 9.23 9.23

3 17.2 17.2

4 17.3 17.4

5

Load Load Load

Case 2 Case 3 Case 4

Frequency Frequency Frequency Maximum

(Hz) (Hz) (Hz) Difference

(%)9.19 9.20 9.22 0.3

9.23 9.25 9.30 0.8

17.2 17.2 17.3 0.6

17.4 17.4 17.4 0.6

29.5 29.6 29.7 0.729.5 29.5

30.4 30.46 30.4 30.5 30.6 0.7

Note: * Difference (%) = (fioaa-fnoload '/fnoload * 100

108

Table 3

Transport Model

Mode Radius Comparison

Mode

No Load

Radius

(Inch)

Load

Case 1

Radius

(Inch)

Load

Case 2

Radius

(Inch)

Load

Case 3

Radius

(Inch)

Load

Case 4

Radius

(Inch)

Maximum

Difference

(%)*39.4

Note:

39.7

39.639.5 39.4

39.8 39.7

7.68 7.67

8.24 8.20

-3.66 -3.67

-3.18 -3.17

40.0

40.2

41.1

1.8

3.5

3 7.68 7.72 7.87 2.5

4 8.21 8.28 8.54 4.0

5 -3.67 -3.67 -3.64 -0.8

6 -3.17 -3.15 -3.14 -0.9

* Difference (%) = (Rload-R.olo_)/R.olo.d * 100

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